Systems of Equations and Inequalities

# Systems of Equations and Inequalities

Figure 1 enigma machines like this one, once owned by Italian dictator benito mussolini, were used by government and military officials for enciphering and deciphering top-secret communications during World War II. (credit: dave Addey, Flickr)

Introduction By 1943, it was obvious to the Nazi regime that defeat was imminent unless it could build a weapon with unlimited destructive power, one that had never been seen before in the history of the world. In September, Adolf Hitler ordered German scientists to begin building an atomic bomb. Rumors and whispers began to spread from across the ocean. Refugees and diplomats told of the experiments happening in Norway. However, Franklin D. Roosevelt wasn’t sold, and even doubted British Prime Minister Winston Churchill’s warning. Roosevelt wanted undeniable proof. Fortunately, he soon received the proof he wanted when a group of mathematicians cracked the “Enigma” code, proving beyond a doubt that Hitler was building an atomic bomb. The next day, Roosevelt gave the order that the United States begin work on the same.

The Enigma is perhaps the most famous cryptographic device ever known. It stands as an example of the pivotal role cryptography has played in society. Now, technology has moved cryptanalysis to the digital world.

Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a matrix is generally part of the process of decoding. In addition to knowing the matrix and its inverse, the receiver must also know the key that, when used with the matrix inverse, will allow the message to be read.

In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.

ChAPTeR OUTlIne

11.1 Systems of linear equations: Two variables 11.2 Systems of linear equations: Three variables 11.3 Systems of nonlinear equations and Inequalities: Two variables 11.4 Partial Fractions 11.5 matrices and matrix Operations 11.6 Solving Systems with gaussian elimination 11.7 Solving Systems with Inverses 11.8 Solving Systems with Cramer’s Rule

SECTION 11.1 sectioN exercises 889

11.1 SeCTIOn exeRCISeS

veRbAl

1. Can a system of linear equations have exactly two solutions? Explain why or why not.

2. If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company’s profit margins.

3. If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?

4. If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?

5. Given a system of equations, explain at least two different methods of solving that system.

AlgebRAIC

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.

6. 5x − y = 4 x + 6y = 2 and (4, 0)

7. −3x − 5y = 13 − x + 4y = 10 and (−6, 1)

8. 3x + 7y = 1 2x + 4y = 0 and (2, 3)

9. −2x + 5y = 7 2x + 9y = 7 and (−1, 1)

10. x + 8y = 43 3x − 2y = −1 and (3, 5)

For the following exercises, solve each system by substitution.

11. x + 3y = 5 2x + 3y = 4

12. 3x − 2y = 18 5x + 10y = −10

13. 4x + 2y = −10 3x + 9y = 0

14. 2x + 4y = −3.8 9x − 5y = 1.3

15. −2x + 3y = 1.2 −3x − 6y = 1.8

16. x − 0.2y = 1 −10x + 2y = 5

17. 3 x + 5y = 9 30x + 50y = −90

18. −3x + y = 2 12x − 4y = −8

19. 1 __ 2 x + 1 __ 3 y = 16

1 __ 6 x + 1 __ 4 y = 9

20. − 1 __ 4 x + 3 __ 2 y = 11

− 1 __ 8 x + 1 __ 3 y = 3

For the following exercises, solve each system by addition.

21. −2x + 5y = −42 7x + 2y = 30

22. 6x − 5y = −34 2x + 6y = 4

23. 5x − y = −2.6 −4x − 6y = 1.4

24. 7x − 2y = 3 4x + 5y = 3.25

25. −x + 2y = −1 5x − 10y = 6

26. 7x + 6y = 2 −28x − 24y = −8

27. 5 __ 6 x + 1 __ 4 y = 0

1 __ 8 x − 1 __ 2 y = −

43 ___ 120

28. 1 __ 3 x + 1 __ 9 y =

2 __ 9

− 1 __ 2 x + 4 __ 5 y = −

1 __ 3

29. −0.2x + 0.4y = 0.6 x − 2y = −3

30. −0.1x + 0.2y = 0.6 5x − 10y = 1

For the following exercises, solve each system by any method.

31. 5x + 9y = 16 x + 2y = 4

32. 6x − 8y = −0.6 3x + 2y = 0.9

33. 5x − 2y = 2.25 7x − 4y = 3

34. x − 5 ___ 12 y = − 55 ___ 12

−6x + 5 __ 2 y = 55 ___ 2

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890 CHAPTER 11 systems of equatioNs aNd iNequalities

35. 7x − 4y = 7 __ 6

2x + 4y = 1 __ 3

36. 3x + 6y = 11 2x + 4y = 9

37. 7 __ 3 x − 1 __ 6 y = 2

− 21 ___ 6 x + 3 ___ 12 y = −3

38. 1 __ 2 x + 1 __ 3 y =

1 __ 3

3 __ 2 x + 1 __ 4 y = −

1 __ 8

39. 2.2x + 1.3y = −0.1 4.2x + 4.2y = 2.1

40. 0.1x + 0.2y = 2 0.35x − 0.3y = 0

gRAPhICAl

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

41. 3x − y = 0.6 x − 2y = 1.3

42. −x + 2y = 4 2x − 4y = 1

43. x + 2y = 7 2x + 6y = 12

44. 3x − 5y = 7 x − 2y = 3

45. 3x − 2y = 5 −9x + 6y = −15

TeChnOlOgy

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

46. 0.1x + 0.2y = 0.3 −0.3x + 0.5y = 1

47. −0.01x + 0.12y = 0.62 0.15x + 0.20y = 0.52

48. 0.5x + 0.3y = 4 0.25x − 0.9y = 0.46

49. 0.15x + 0.27y = 0.39 −0.34x + 0.56y = 1.8

50. −0.71x + 0.92y = 0.13 0.83x + 0.05y = 2.1

exTenSIOnS

For the following exercises, solve each system in terms of A, B, C, D, E, and F where A – F are nonzero numbers. Note that A ≠ B and AE ≠ BD.

51. x + y = A x − y = B

52. x + Ay = 1 x + By = 1

53. Ax + y = 0 Bx + y = 1

54. Ax + By = C x + y = 1

55. Ax + By = C Dx + Ey = F

ReAl-WORld APPlICATIOnS

For the following exercises, solve for the desired quantity.

56. A stuffed animal business has a total cost of production C = 12x + 30 and a revenue function R = 20x. Find the break-even point.

57. A fast-food restaurant has a cost of production C(x) = 11x + 120 and a revenue function R(x) = 5x. When does the company start to turn a profit?

58. A cell phone factory has a cost of production C(x) = 150x + 10,000 and a revenue function R(x) = 200x. What is the break-even point?

59. A musician charges C(x) = 64x + 20,000, where x is the total number of attendees at the concert. The venue charges \$80 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

60. A guitar factory has a cost of production C(x) = 75x + 50,000. If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

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SECTION 11.1 sectioN exercises 891

For the following exercises, use a system of linear equations with two variables and two equations to solve.

61. Find two numbers whose sum is 28 and difference is 13.

62. A number is 9 more than another number. Twice the sum of the two numbers is 10. Find the two numbers.

63. The startup cost for a restaurant is \$120,000, and each meal costs \$10 for the restaurant to make. If each meal is then sold for \$15, after how many meals does the restaurant break even?

64. A moving company charges a flat rate of \$150, and an additional \$5 for each box. If a taxi service would charge \$20 for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

65. A total of 1,595 first- and second-year college students gathered at a pep rally. The number of freshmen exceeded the number of sophomores by 15. How many freshmen and sophomores were in attendance?

66. 276 students enrolled in a freshman-level chemistry class. By the end of the semester, 5 times the number of students passed as failed. Find the number of students who passed, and the number of students who failed.

67. There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?

68. A jeep and BMW enter a highway running east- west at the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 7 mph slower than the BMW. After 2 hours from the time the BMW entered the highway, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.

69. If a scientist mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many gallons of 10% and 60% solutions were mixed?

70. An investor earned triple the profits of what she earned last year. If she made \$500,000.48 total for both years, how much did she earn in profits each year?

71. An investor who dabbles in real estate invested 1.1 million dollars into two land investments. On the first investment, Swan Peak, her return was a 110% increase on the money she invested. On the second investment, Riverside Community, she earned 50% over what she invested. If she earned \$1 million in profits, how much did she invest in each of the land deals?

72. If an investor invests a total of \$25,000 into two bonds, one that pays 3% simple interest, and the

other that pays 2 7 __ 8 % interest, and the investor

earns \$737.50 annual interest, how much was invested in each account?

73. If an investor invests \$23,000 into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns \$710.00 annual interest, how much was invested in each account?

74. CDs cost \$5.96 more than DVDs at All Bets Are Off Electronics. How much would 6 CDs and 2 DVDs cost if 5 CDs and 2 DVDs cost \$127.73?

75. A store clerk sold 60 pairs of sneakers. The high-tops sold for \$98.99 and the low-tops sold for \$129.99. If the receipts for the two types of sales totaled \$6,404.40, how many of each type of sneaker were sold?

76. A concert manager counted 350 ticket receipts the day after a concert. The price for a student ticket was \$12.50, and the price for an adult ticket was \$16.00. The register confirms that \$5,075 was taken in. How many student tickets and adult tickets were sold?

77. Admission into an amusement park for 4 children and 2 adults is \$116.90. For 6 children and 3 adults, the admission is \$175.35. Assuming a different price for children and adults, what is the price of the child’s ticket and the price of the adult ticket?

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SECTION 11.2 sectioN exercises 899

11.2 SeCTIOn exeRCISeS

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1. Can a linear system of three equations have exactly two solutions? Explain why or why not

2. If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.

3. If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.

4. Using the method of addition, is there only one way to solve the system?

5. Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.

AlgebRAIC

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.

6. 2x − 6y + 6z = −12 x + 4y + 5z = −1 and (0, 1, −1) −x + 2y + 3z = −1

7. 6x − y + 3z = 6 3x + 5y + 2z = 0 and (3, −3, −5) x + y = 0

8. 6x − 7y + z = 2 −x − y + 3z = 4 and (4, 2, −6) 2x + y − z = 1

9. x − y = 0 x − z = 5 and (4, 4, −1) x − y + z = −1

10. −x − y + 2z = 3 5x + 8y − 3z = 4 and (4, 1, −7) −x + 3y − 5z = −5

For the following exercises, solve each system by substitution.

11. 3x − 4y + 2z = −15 2x + 4y + z = 16 2x + 3y + 5z = 20

12. 5x − 2y + 3z = 20 2x − 4y − 3z = −9 x + 6y − 8z = 21

13. 5x + 2y + 4z = 9 −3x + 2y + z = 10 4x − 3y + 5z = −3

14. 4x − 3y + 5z = 31 −x + 2y + 4z = 20 x + 5y − 2z = −29

15. 5x − 2y + 3z = 4 −4x + 6y − 7z = −1 3x + 2y − z = 4

16. 4x + 6y + 9z = 0 −5x + 2y − 6z = 3 7x − 4y + 3z = −3

For the following exercises, solve each system by Gaussian elimination.

17. 2x − y + 3z = 17 −5x + 4y − 2z = −46 2y + 5z = −7

18. 5x − 6y + 3z = 50 − x + 4y = 10 2x − z = 10

19. 2x + 3y − 6z = 1 −4x − 6y + 12z = −2 x + 2y + 5z = 10

20. 4x + 6y − 2z = 8 6x + 9y − 3z = 12 −2x − 3y + z = −4

21. 2x + 3y − 4z = 5 −3x + 2y + z = 11 −x + 5y + 3z = 4

22. 10x + 2y − 14z = 8 −x − 2y − 4z = −1 −12x − 6y + 6z = −12

23. x + y + z = 14 2y + 3z = −14 −16y − 24z = −112

24. 5x − 3y + 4z = −1 −4x + 2y − 3z = 0 −x + 5y + 7z = −11

25. x + y + z = 0 2x − y + 3z = 0 x − z = 0

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900 CHAPTER 11 systems of equatioNs aNd iNequalities

26. 3x + 2y − 5z = 6 5x − 4y + 3z = −12 4x + 5y−2z = 15

27. x + y + z = 0 2x − y + 3z = 0 x − z = 1

28. 3x − 1 __ 2 y − z = − 1 __ 2

4x + z = 3

−x + 3 __ 2 y = 5 __ 2

29. 6x − 5y + 6z = 38

1 __ 5 x − 1 __ 2 y +

3 __ 5 z = 1

−4x − 3 __ 2 y − z = −74

30. 1 __ 2 x − 1 __ 5 y +

2 __ 5 z = − 13 ___ 10

1 __ 4 x − 2 __ 5 y −

1 __ 5 z = − 7 ___ 20

− 1 __ 2 x − 3 __ 4 y −

1 __ 2 z = − 5 __ 4

31. − 1 __ 3 x − 1 __ 2 y −

1 __ 4 z = 3 __ 4

− 1 __ 2 x − 1 __ 4 y −

1 __ 2 z = 2

− 1 __ 4 x − 3 __ 4 y −

1 __ 2 z = − 1 __ 2

32. 1 __ 2 x − 1 __ 4 y +

3 __ 4 z = 0

1 __ 4 x − 1 ___ 10 y +

2 __ 5 z = −2

1 __ 8 x + 1 __ 5 y −

1 __ 8 z = 2

33. 4 __ 5 x − 7 __ 8 y +

1 __ 2 z = 1

− 4 __ 5 x − 3 __ 4 y +

1 __ 3 z = −8

− 2 __ 5 x − 7 __ 8 y +

1 __ 2 z = −5

34. − 1 __ 3 x − 1 __ 8 y +

1 __ 6 z = − 4 __ 3

− 2 __ 3 x − 7 __ 8 y +

1 __ 3 z = − 23 ___ 3

− 1 __ 3 x − 5 __ 8 y +

5 __ 6 z = 0

35. − 1 __ 4 x − 5 __ 4 y +

5 __ 2 z = −5

− 1 __ 2 x − 5 __ 3 y +

5 __ 4 z = 55 ___ 12

− 1 __ 3 x − 1 __ 3 y +

1 __ 3 z = 5 __ 3

36. 1 ___ 40 x + 1 ___ 60 y +

1 ___ 80 z = 1 ___ 100

− 1 __ 2 x − 1 __ 3 y −

1 __ 4 z = − 1 __ 5

3 __ 8 x + 3 ___ 12 y +

3 ___ 16 z = 3 ___ 20

37. 0.1x − 0.2y + 0.3z = 2 0.5x − 0.1y + 0.4z = 8 0.7x − 0.2y + 0.3z = 8

38. 0.2x + 0.1y − 0.3z = 0.2 0.8x + 0.4y − 1.2z = 0.1 1.6x + 0.8y − 2.4z = 0.2

39. 1.1x + 0.7y − 3.1z = −1.79 2.1x + 0.5y − 1.6z = −0.13 0.5x + 0.4y − 0.5z = −0.07

40. 0.5x − 0.5y + 0.5z = 10 0.2x − 0.2y + 0.2z = 4 0.1x − 0.1y + 0.1z = 2

41. 0.1x + 0.2y + 0.3z = 0.37 0.1x − 0.2y − 0.3z = −0.27 0.5x − 0.1y − 0.3z = −0.03

42. 0.5x − 0.5y − 0.3z = 0.13 0.4x − 0.1y − 0.3z = 0.11 0.2x − 0.8y − 0.9z = −0.32

43. 0.5x + 0.2y − 0.3z = 1 0.4x − 0.6y + 0.7z = 0.8 0.3x − 0.1y − 0.9z = 0.6

44. 0.3x + 0.3y + 0.5z = 0.6 0.4x + 0.4y + 0.4z = 1.8 0.4x + 0.2y + 0.1z = 1.6

45. 0.8x + 0.8y + 0.8z = 2.4 0.3x − 0.5y + 0.2z = 0 0.1x + 0.2y + 0.3z = 0.6

exTenSIOnS

For the following exercises, solve the system for x, y, and z.

46. x + y + z = 3

x − 1 _____ 2 + y − 3

_____ 2 + z + 1 _____ 2 = 0

x − 2 _____ 3 + y + 4 _____ 3 +

z − 3 _____ 3 = 2 __ 3

47. 5x − 3y − z + 1 _____ 2 = 1 __ 2

6x + y − 9

_____ 2 + 2z = −3

x + 8 _____ 2 − 4y + z = 4

48. x + 4 _____ 7 − y − 1

_____ 6 + z + 2 _____ 3 = 1

x − 2 _____ 4 + y + 1

_____ 8 − z + 8 _____ 12 = 0

x + 6 _____ 3 − y + 2

_____ 3 + z + 4 _____ 2 = 3

49. x − 3 _____ 6 + y + 2

_____ 2 − z − 3 _____ 3 = 2

x + 2 _____ 4 + y − 5

_____ 2 + z + 4 _____ 2 = 1

x + 6 _____ 2 − y − 3

_____ 2 + z + 1 = 9

50. x − 1 ____ 3 + y + 3

_____ 4 + z + 2 _____ 6 = 1

4x + 3y − 2z = 11

0.02x + 0.015y − 0.01z = 0.065

SECTION 11.2 sectioN exercises 901

ReAl-WORld APPlICATIOnS

51. Three even numbers sum up to 108. The smaller is half the larger and the middle number is 3 _ 4 the larger. What are the three numbers?

52. Three numbers sum up to 147. The smallest number is half the middle number, which is half the largest number. What are the three numbers?

53. At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?

54. An animal shelter has a total of 350 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than one-half the number of cats, and there are 20 more cats than dogs, how many of each animal are at the shelter?

55. Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was \$82. She forgot to save the individual receipts but remembered that your groceries were \$0.05 cheaper than half of her groceries, and that your other roommate’s groceries were \$2.10 more than your groceries. How much was each of your share of the groceries?

56. Your roommate, John, offered to buy household supplies for you and your other roommate. You live near the border of three states, each of which has a different sales tax. The total amount of money spent was \$100.75. Your supplies were bought with 5% tax, John’s with 8% tax, and your third roommate’s with 9% sales tax. The total amount of money spent without taxes is \$93.50. If your supplies before tax were \$1 more than half of what your third roommate’s supplies were before tax, how much did each of you spend? Give your answer both with and without taxes.

57. Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is \$82,000. The office manager makes \$4,000 more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total \$78,000. What is the annual salary of each of the co-workers?

58. At a carnival, \$2,914.25 in receipts were taken at the end of the day. The cost of a child’s ticket was \$20.50, an adult ticket was \$29.75, and a senior citizen ticket was \$15.25. There were twice as many senior citizens as adults in attendance, and 20 more children than senior citizens. How many children, adult, and senior citizen tickets were sold?

59. A local band sells out for their concert. They sell all 1,175 tickets for a total purse of \$28,112.50. The tickets were priced at \$20 for student tickets, \$22.50 for children, and \$29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?

60. In a bag, a child has 325 coins worth \$19.50. There were three types of coins: pennies, nickels, and dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was in the bag?

61. Last year, at Haven’s Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of \$140,000. This year, due to inflation, the same cars would cost \$151,830. The cost of the BMW increased by 8%, the Jeep by 5%, and the Toyota by 12%. If the price of last year’s Jeep was \$7,000 less than the price of last year’s BMW, what was the price of each of the three cars last year?

62. A recent college graduate took advantage of his business education and invested in three investments immediately after graduating. He invested \$80,500 into three accounts, one that paid 4% simple interest, one that paid 4% simple interest, one that paid 3 1 __ 8 % simple interest, and one that paid 2 1 __ 2 % simple interest. He earned \$2,670 interest at the end of one year. If the amount of the money invested in the second account was four times the amount invested in the third account, how much was invested in each account?

902 CHAPTER 11 systems of equatioNs aNd iNequalities

63. You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3% compounded annually, the second account pays 4% compounded annually, and the third account pays 2% compounded annually. After one year, you earn \$34,000 in interest. If you invest four times the money into the account that pays 3% compared to 2%, how much did you invest in each account?

64. You inherit one hundred thousand dollars. You invest it all in three accounts for one year. The first account pays 4% compounded annually, the second account pays 3% compounded annually, and the third account pays 2% compounded annually. After one year, you earn \$3,650 in interest. If you invest five times the money in the account that pays 4% compared to 3%, how much did you invest in each account?

65. The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed 39.8% of the world’s consumed oil. The United States consumed 0.7% more than four times China’s consumption. The United States consumed 5% more than triple Japan’s consumption. What percent of the world oil consumption did the United States, Japan, and China consume?[28]

66. The top three countries in oil production in the same year are Saudi Arabia, the United States, and Russia. In millions of barrels per day, the top three countries produced 31.4% of the world’s produced oil. Saudi Arabia and the United States combined for 22.1% of the world’s production, and Saudi Arabia produced 2% more oil than Russia. What percent of the world oil production did Saudi Arabia, the United States, and Russia produce?[29]

67. The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexico, and Canada. The three top countries accounted for 47% of oil imports. The United States imported 1.8% more from Saudi Arabia than they did from Mexico, and 1.7% more from Saudi Arabia than they did from Canada. What percent of the United States oil imports were from these three countries?[30]

68. The top three oil producers in the United States in a certain year are the Gulf of Mexico, Texas, and Alaska. The three regions were responsible for 64% of the United States oil production. The Gulf of Mexico and Texas combined for 47% of oil production. Texas produced 3% more than Alaska. What percent of United States oil production came from these regions?[31]

69. At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55% of the endangered species. Birds accounted for 0.7% more than fish, and fish accounted for 1.5% more than mammals. What percent of the endangered species came from mammals, birds, and fish?

70. Meat consumption in the United States can be broken into three categories: red meat, poultry, and fish. If fish makes up 4% less than one-quarter of poultry consumption, and red meat consumption is 18.2% higher than poultry consumption, what are the percentages of meat consumption?[32]

28 “Oil reserves, production and consumption in 2001,” accessed April 6, 2014, http://scaruffi.com/politics/oil.html. 29 “Oil reserves, production and consumption in 2001,” accessed April 6, 2014, http://scaruffi.com/politics/oil.html. 30 “Oil reserves, production and consumption in 2001,” accessed April 6, 2014, http://scaruffi.com/politics/oil.html. 31 “USA: The coming global oil crisis,” accessed April 6, 2014, http://www.oilcrisis.com/us/. 32 “The United States Meat Industry at a Glance,” accessed April 6, 2014, http://www.meatami.com/ht/d/sp/i/47465/pid/ 47465.

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944 CHAPTER 11 systems of equatioNs aNd iNequalities

11.6 SeCTIOn exeRCISeS

veRbAl

1. Can any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.

2. Can any matrix be written as a system of linear equations? Explain why or why not. Explain how to write that system of equations.

3. Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the augmented matrix

 9 3 1 −2 ∣ 0 6  . 4. Can a matrix whose entry is 0 on the diagonal be

solved? Explain why or why not. What would you do to remedy the situation?

5. Can a matrix that has 0 entries for an entire row have one solution? Explain why or why not.

AlgebRAIC

For the following exercises, write the augmented matrix for the linear system.

6. 8x − 37y = 8 2x + 12y = 3

7. 16y = 4 9x − y = 2

8. 3x + 2y + 10z = 3 −6x + 2y + 5z = 13 4x + z = 18

9. x + 5y + 8z = 19 12x + 3y = 4 3x + 4y + 9z = −7

10. 6x + 12y + 16z = 4 19x − 5y + 3z = −9 x + 2y = −8

For the following exercises, write the linear system from the augmented matrix.

11.  −2 5 6 −18 ∣ 5 26  12.  3 4 10 17 ∣ 10 439  13.  3 2 0

−1 −9 4 8 5 7

∣ 3 −1 8  14. 

8 29 1

−1 7 5

0 0 3 ∣ 43 38 10  15.  4 5 −2 0 1 58 8 7 −3 ∣ 12 2 −5 

For the following exercises, solve the system by Gaussian elimination.

16.  1 0 0 0 ∣ 3 0  17.  1 0 1 0 ∣ 1 2  18.  1 2 4 5 ∣ 3 6  19.  −1 2 4 −5 ∣ −3 6  20.  −2 0 0 2 ∣ 1 −1  21. 2x − 3y = − 9 5x + 4y = 58 22. 6x + 2y = −4 3x + 4y = −17 23. 2x + 3y = 12 4x + y = 14 24. −4x − 3y = −2

3x − 5y = −13 25. −5x + 8y = 3

10x + 6y = 5 26. 3x + 4y = 12

−6x − 8y = −24 27. −60x + 45y = 12

20x − 15y = −4

28. 11x + 10y = 43 15x + 20y = 65

29. 2x − y = 2 3x + 2y = 17

30. −1.06x−2.25y = 5.51 −5.03x − 1.08y = 5.40

31. 3 __ 4 x − 3 __ 5 y = 4

1 __ 4 x + 2 __ 3 y = 1

32. 1 __ 4 x − 2 __ 3 y = −1

1 __ 2 x + 1 __ 3 y = 3

33. 

1 0 0

0 1 1

0 0 1 ∣ 31 45 87  34.  1 0 1 1 1 0 0 1 1 ∣ 50 20 −90  35.  1 2 3 0 5 6 0 0 8 ∣ 4 7 9 

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SECTION 11.6 sectioN exercises 945

36. 

−0.1 0.3 −0.1

−0.4 0.2 0.1

0.6 0.1 0.7 ∣ 0.2 0.8 −0.8  37. −2x + 3y − 2z = 3 4x + 2y − z = 9 4x − 8y + 2z = −6 38. x + y − 4z = −4 5x − 3y − 2z = 0 2x + 6y + 7z = 30

39. 2x + 3y + 2z = 1 −4x − 6y − 4z = −2 10x + 15y + 10z = 5

40. x + 2y − z = 1 −x − 2y + 2z = −2 3x + 6y − 3z = 5

41. x + 2y − z = 1 −x − 2y + 2z = −2 3x + 6y − 3z = 3

42. x + y = 2 x + z = 1 −y − z = −3

43. x + y + z = 100 x + 2z = 125 −y + 2z = 25

44. 1 __ 4 x − 2 __ 3 z = −

1 __ 2

1 __ 5 x + 1 __ 3 y =

4 __ 7

1 __ 5 y − 1 __ 3 z =

2 __ 9

45. − 1 __ 2 x + 1 __ 2 y +

1 __ 7 z = − 53 ___ 14

1 __ 2 x − 1 __ 2 y +

1 __ 4 z = 3

1 __ 4 x + 1 __ 5 y +

1 __ 3 z = 23 ___ 15

46. − 1 __ 2 x − 1 __ 3 y +

1 __ 4 z = − 29 ___ 6

1 __ 5 x + 1 __ 6 y −

1 __ 7 z = 431 ___ 210

− 1 __ 8 x + 1 __ 9 y +

1 ___ 10 z = − 49 ___ 45

exTenSIOnS

For the following exercises, use Gaussian elimination to solve the system.

47. x − 1 _____ 7 + y − 2

_____ 8 + z − 3 _____ 4 = 0

x + y + z = 6

x + 2 _____ 3 + 2y + z−3 ____ 3 = 5

48. x − 1 _____ 4 − y + 1

_____ 4 + 3z = −1

x + 5 _____ 2 + y + 7

_____ 4 − z = 4

x + y − z−2 ____ 2 = 1

49. x − 3 _____ 4 − y − 1

_____ 3 + 2z = −1

x + 5 _____ 2 + y + 5

_____ 2 + z + 5 _____ 2 = 8

x + y + z = 1

50. x − 3 _____ 10 + y + 3

_____ 2 −2z = 3

x + 5 _____ 4 − y − 1

_____ 8 + z = 3 __ 2

x − 1 _____ 4 + y + 4

_____ 2 + 3z = 3 __ 2

51. x − 3 _____ 4 − y − 1

_____ 3 + 2z = −1

x + 5 _____ 2 + y + 5

_____ 2 + z + 5 _____ 2 = 7

x + y + z = 1

ReAl-WORld APPlICATIOnS

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.

52. Every day, a cupcake store sells 5,000 cupcakes in chocolate and vanilla flavors. If the chocolate flavor is 3 times as popular as the vanilla flavor, how many of each cupcake sell per day?

53. At a competing cupcake store, \$4,520 worth of cupcakes are sold daily. The chocolate cupcakes cost \$2.25 and the red velvet cupcakes cost \$1.75. If the total number of cupcakes sold per day is 2,200, how many of each flavor are sold each day?

54. You invested \$10,000 into two accounts: one that has simple 3% interest, the other with 2.5% interest. If your total interest payment after one year was \$283.50, how much was in each account after the year passed?

55. You invested \$2,300 into account 1, and \$2,700 into account 2. If the total amount of interest after one year is \$254, and account 2 has 1.5 times the interest rate of account 1, what are the interest rates? Assume simple interest rates.

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946 CHAPTER 11 systems of equatioNs aNd iNequalities

56. Bikes’R’Us manufactures bikes, which sell for \$250. It costs the manufacturer \$180 per bike, plus a startup fee of \$3,500. After how many bikes sold will the manufacturer break even?

57. A major appliance store is considering purchasing vacuums from a small manufacturer. The store would be able to purchase the vacuums for \$86 each, with a delivery fee of \$9,200, regardless of how many vacuums are sold. If the store needs to start seeing a profit after 230 units are sold, how much should they charge for the vacuums?

58. The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising 83% of the flavors sold at an ice cream shop. If vanilla sells 1% more than twice strawberry, and chocolate sells 11% more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

59. At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up 12% of total ice cream sales. This year, the same three ice creams made up 16.9% of ice cream sales. The rocky road sales doubled, the banana sales increased by 50%, and the pumpkin sales increased by 20%. If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year.

60. A bag of mixed nuts contains cashews, pistachios, and almonds. There are 1,000 total nuts in the bag, and there are 100 less almonds than pistachios. The cashews weigh 3 g, pistachios weigh 4 g, and almonds weigh 5 g. If the bag weighs 3.7 kg, find out how many of each type of nut is in the bag.

61. A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. 30% of the almonds, 20% of the cashews, and 10% of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

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972 CHAPTER 11 systems of equatioNs aNd iNequalities

ChAPTeR 11 RevIeW

Key Terms addition method an algebraic technique used to solve systems of linear equations in which the equations are added in a

way that eliminates one variable, allowing the resulting equation to be solved for the remaining variable; substitution is then used to solve for the first variable

augmented matrix a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets break-even point the point at which a cost function intersects a revenue function; where profit is zero coefficient matrix a matrix that contains only the coefficients from a system of equations column a set of numbers aligned vertically in a matrix consistent system a system for which there is a single solution to all equations in the system and it is an independent system,

or if there are an infinite number of solutions and it is a dependent system cost function the function used to calculate the costs of doing business; it usually has two parts, fixed costs and variable costs Cramer’s Rule a method for solving systems of equations that have the same number of equations as variables using determinants dependent system a system of linear equations in which the two equations represent the same line; there are an infinite

number of solutions to a dependent system determinant a number calculated using the entries of a square matrix that determines such information as whether there

is a solution to a system of equations entry an element, coefficient, or constant in a matrix feasible region the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions

of each inequality intersect Gaussian elimination using elementary row operations to obtain a matrix in row-echelon form identity matrix a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix

algebra inconsistent system a system of linear equations with no common solution because they represent parallel lines, which have

no point or line in common independent system a system of linear equations with exactly one solution pair (x, y) main diagonal entries from the upper left corner diagonally to the lower right corner of a square matrix

matrix a rectangular array of numbers multiplicative inverse of a matrix a matrix that, when multiplied by the original, equals the identity matrix

nonlinear inequality an inequality containing a nonlinear expression partial fraction decomposition the process of returning a simplified rational expression to its original form, a sum or

difference of simpler rational expressions partial fractions the individual fractions that make up the sum or difference of a rational expression before combining them

into a simplified rational expression profit function the profit function is written as P(x) = R(x) − C(x), revenue minus cost revenue function the function that is used to calculate revenue, simply written as R = xp, where x = quantity and p = price row a set of numbers aligned horizontally in a matrix row operations adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the

goal of achieving row-echelon form row-echelon form after performing row operations, the matrix form that contains ones down the main diagonal and zeros

at every space below the diagonal row-equivalent two matrices A and B are row-equivalent if one can be obtained from the other by performing basic row

operations scalar multiple an entry of a matrix that has been multiplied by a scalar solution set the set of all ordered pairs or triples that satisfy all equations in a system of equations substitution method an algebraic technique used to solve systems of linear equations in which one of the two equations is

solved for one variable and then substituted into the second equation to solve for the second variable

CHAPTER 11 review 973

system of linear equations a set of two or more equations in two or more variables that must be considered simultaneously. system of nonlinear equations a system of equations containing at least one equation that is of degree larger than one system of nonlinear inequalities a system of two or more inequalities in two or more variables containing at least one

inequality that is not linear

Key equations Identity matrix for a 2 × 2 matrix I 2 =  1 0 0 1 

Identity matrix for a 3 × 3 matrix I 3 =  1 0 0

0 1 0 0 0 1

 Multiplicative inverse of a 2 × 2 matrix A −1 = 1 _______

ad − bc  d −b −c a  , where ad − bc ≠ 0

Key Concepts

11.1 Systems of Linear Equations: Two Variables • A system of linear equations consists of two or more equations made up of two or more variables such that all equations

in the system are considered simultaneously. • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

See Example 1. • Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions,

or inconsistent with no solution. • One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the

equations on the same set of axes. See Example 2. • Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in

one equation and substitute the result into the second equation. See Example 3. • A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding

opposite coefficients of corresponding variables. See Example 4. • It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding

the two equations together. See Example 5, Example 6, and Example 7. • Either method of solving a system of equations results in a false statement for inconsistent systems because they are

made up of parallel lines that never intersect. See Example 8. • The solution to a system of dependent equations will always be true because both equations describe the same line.

See Example 9. • Systems of equations can be used to solve real-world problems that involve more than one variable, such as those

relating to revenue, cost, and profit. See Example 10 and Example 11.

11.2 Systems of Linear Equations: Three Variables • A solution set is an ordered triple {(x, y, z)} that represents the intersection of three planes in space. See Example 1. • A system of three equations in three variables can be solved by using a series of steps that forces a variable to be

eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See Example 2.

• Systems of three equations in three variables are useful for solving many different types of real-world problems. See Example 3.

• A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See Example 4.

• Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.

974 CHAPTER 11 systems of equatioNs aNd iNequalities

• A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See Example 5.

• Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

11.3 Systems of Nonlinear Equations and Inequalities: Two Variables • There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution,

the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points. See Example 1.

• There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the parabola; (3) two solutions, the line intersects the circle in two points. See Example 2.

• There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle: (1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and ellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points. See Example 3.

• An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade the region containing the solution set. See Example 4.

• Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities. See Example 5.

11.4 Partial Fractions

• Decompose P(x) ____ Q(x)

by writing the partial fractions as A ________ a 1 x + b 1

+ B ________ a 2 x + b 2

. Solve by clearing the fractions, expanding

the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. See Example 1.

• The decomposition of P(x) ____ Q(x)

with repeated linear factors must account for the factors of the denominator in

increasing powers. See Example 2.

• The decomposition of P(x) ____ Q(x)

with a nonrepeated irreducible quadratic factor needs a linear numerator over the

quadratic factor, as in A __ x + Bx + C ____________

(a x 2 + bx + c) . See Example 3.

• In the decomposition of P(x) ____ Q(x)

, where Q(x) has a repeated irreducible quadratic factor, when the irreducible

quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as

Ax + B

__ (a x 2 + bx + c)

+ A 2 x + B 2 __

(a x 2 + bx + c) 2 + … +

A n x + B n __ (a x 2 + bx + c) n

. See Example 4.

11.5 Matrices and Matrix Operations • A matrix is a rectangular array of numbers. Entries are arranged in rows and columns. • The dimensions of a matrix refer to the number of rows and the number of columns. A 3 × 2 matrix has three rows

and two columns. See Example 1. • We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix.

See Example 2, Example 3, Example 4, and Example 5. • Scalar multiplication involves multiplying each entry in a matrix by a constant. See Example 6. • Scalar multiplication is often required before addition or subtraction can occur. See Example 7. • Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must

match the number of rows in the second. • The product of two matrices, A and B, is obtained by multiplying each entry in row 1 of A by each entry in column 1

of B; then multiply each entry of row 1 of A by each entry in columns 2 of B, and so on. See Example 8 and Example 9.

CHAPTER 11 review 975

• Many real-world problems can often be solved using matrices. See Example 10. • We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See Example 11.

11.6 Solving Systems with Gaussian Elimination • An augmented matrix is one that contains the coefficients and constants of a system of equations. See Example 1. • A matrix augmented with the constant column can be represented as the original system of equations. See Example 2. • Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows. • We can use Gaussian elimination to solve a system of equations. See Example 3, Example 4, and Example 5. • Row operations are performed on matrices to obtain row-echelon form. See Example 6. • To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form.

Back-substitute to find the solutions. See Example 7 and Example 8. • A calculator can be used to solve systems of equations using matrices. See Example 9. • Many real-world problems can be solved using augmented matrices. See Example 10 and Example 11.

11.7 Solving Systems with Inverses • An identity matrix has the property AI = IA = A. See Example 1. • An invertible matrix has the property A A −1 = A −1 A = I. See Example 2. • Use matrix multiplication and the identity to find the inverse of a 2 × 2 matrix. See Example 3. • The multiplicative inverse can be found using a formula. See Example 4. • Another method of finding the inverse is by augmenting with the identity. See Example 5. • We can augment a 3 × 3 matrix with the identity on the right and use row operations to turn the original matrix into

the identity, and the matrix on the right becomes the inverse. See Example 6. • Write the system of equations as AX = B, and multiply both sides by the inverse of A: A −1 AX = A −1 B. See Example 7

and Example 8. • We can also use a calculator to solve a system of equations with matrix inverses. See Example 9.

11.8 Solving Systems with Cramer’s Rule

• The determinant for  a b c d  is ad − bc. See Example 1. • Cramer’s Rule replaces a variable column with the constant column. Solutions are x =

D x _ D

, y = D y

_ D

. See Example 2.

• To find the determinant of a 3 × 3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See Example 3.

• To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant

column for each desired solution: x = D x _ D

, y = D y

_ D

, z = D z _ D

. See Example 4.

• Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See Example 5 and Example 6.

• Certain properties of determinants are useful for solving problems. For example:

○ If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.

○ When two rows are interchanged, the determinant changes sign. ○ If either two rows or two columns are identical, the determinant equals zero. ○ If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero. ○ The determinant of an inverse matrix A −1 is the reciprocal of the determinant of the matrix A. ○ If any row or column is multiplied by a constant, the determinant is multiplied by the same factor. See

Example 7 and Example 8.

976 CHAPTER 11 systems of equatioNs aNd iNequalities

ChAPTeR 11 RevIeW exeRCISeS

SySTemS OF lIneAR eQUATIOnS: TWO vARIAbleS

For the following exercises, determine whether the ordered pair is a solution to the system of equations.

1. 3x − y = 4 x + 4y = − 3 and ( − 1, 1)

2. 6x − 2y = 24 −3x + 3y = 18 and (9, 15)

For the following exercises, use substitution to solve the system of equations.

3. 10x + 5y = −5 3x − 2y = −12

4. 4 __ 7 x + 1 __ 5 y =

43 ___ 70

5 __ 6 x − 1 __ 3 y = −

2 __ 3

5. 5x + 6y = 14 4x + 8y = 8

For the following exercises, use addition to solve the system of equations.

6. 3x + 2y = −7 2x + 4y = 6

7. 3x + 4y = 2 9x + 12y = 3

8. 8x + 4y = 2 6x − 5y = 0.7

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

9. A factory has a cost of production C(x) = 150x + 15,000 and a revenue function R(x) = 200x. What is the break-even point?

10. A performer charges C(x) = 50x + 10,000, where x is the total number of attendees at a show. The venue charges \$75 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

SySTemS OF lIneAR eQUATIOnS: ThRee vARIAbleS

For the following exercises, solve the system of three equations using substitution or addition.

11. 0.5x − 0.5y = 10 − 0.2y + 0.2x = 4 0.1x + 0.1z = 2

12. 5x + 3y − z = 5 3x − 2y + 4z = 13 4x + 3y + 5z = 22

13. x + y + z = 1 2x + 2y + 2z = 1 3x + 3y = 2

14. 2x − 3y + z = −1 x + y + z = −4 4x + 2y − 3z = 33

15. 3x + 2y − z = −10 x − y + 2z = 7 −x + 3y + z = −2

16. 3x + 4z = −11 x − 2y = 5 4y − z = −10

17. 2x − 3y + z = 0 2x + 4y − 3z = 0 6x − 2y − z = 0

18. 6x − 4y − 2z = 2 3x + 2y − 5z = 4 6y − 7z = 5

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

19. Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?

20. A local theatre sells out for their show. They sell all 500 tickets for a total purse of \$8,070.00. The tickets were priced at \$15 for students, \$12 for children, and \$18 for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

CHAPTER 11 review 977

SySTemS OF nOnlIneAR eQUATIOnS And IneQUAlITIeS: TWO vARIAbleS

For the following exercises, solve the system of nonlinear equations.

21. y = x 2 − 7 y = 5x − 13

22. y = x 2 − 4 y = 5x + 10

23. x 2 + y 2 = 16 y = x − 8

24. x 2 + y 2 = 25 y = x 2 + 5

25. x 2 + y 2 = 4 y − x 2 = 3

For the following exercises, graph the inequality.

26. y > x 2 − 1 27. 1 __ 4 x 2 + y 2 < 4

For the following exercises, graph the system of inequalities.

28. x 2 + y 2 + 2x < 3 y > − x 2 − 3

29. x 2 − 2x + y 2 − 4x < 4 y < − x + 4

30. x 2 + y 2 < 1 y 2 < x

PARTIAl FRACTIOnS

For the following exercises, decompose into partial fractions.

31. −2x + 6 __________ x 2 + 3x + 2

32. 10x + 2 ___________ 4 x 2 + 4x + 1

33. 7x + 20 ____________ x 2 + 10x + 25

34. x − 18 ____________ x 2 − 12x + 36

35. − x 2 + 36x + 70 _____________ x 3 − 125

36. −5 x 2 + 6x − 2 ____________

x 3 + 27

37. x 3 − 4 x 2 + 3x + 11 ________________

( x 2 − 2) 2 38. 4 x

4 − 2 x 3 + 22 x 2 − 6x + 48 _______________________ x( x 2 + 4) 2

mATRICeS And mATRIx OPeRATIOnS

For the following exercises, perform the requested operations on the given matrices.

A =  4 −2 1 3  , B =  6 7 −3

11 −2 4  , C = 

6 7

11 −2

14 0  , D = 

1 −4 9

10 5 −7

2 8 5  , E = 

7 −14 3

2 −1 3

0 1 9 

39. −4A 40. 10D − 6E 41. B + C 42. AB 43. BA 44. BC 45. CB 46. DE 47. ED 48. EC 49. CE 50. A 3

SOlvIng SySTemS WITh gAUSSIAn elImInATIOn

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

51.  1 0 −3

0 1 2 0 0 0

∣ 7 −5 0  52.  1 0 5 0 1 −2 0 0 0 ∣ −9 4 3  For the following exercises, write the augmented matrix from the system of linear equations.

53. −2x + 2y + z = 7 2x − 8y + 5z = 0 19x − 10y + 22z = 3

54. 4x + 2y − 3z = 14 −12x + 3y + z = 100 9x − 6y + 2z = 31

55. x + 3z = 12 −x + 4y = 0 y + 2z = − 7

978 CHAPTER 11 systems of equatioNs aNd iNequalities

For the following exercises, solve the system of linear equations using Gaussian elimination.

56. 3x − 4y = − 7 −6x + 8y = 14

57. 3x − 4y = 1 −6x + 8y = 6

58. −1.1x − 2.3y = 6.2 −5.2x − 4.1y = 4.3

59. 2x + 3y + 2z = 1 −4x − 6y − 4z = − 2 10x + 15y + 10z = 0

60. −x + 2y − 4z = 8 3y + 8z = − 4 −7x + y + 2z = 1

SOlvIng SySTemS WITh InveRSeS

For the following exercises, find the inverse of the matrix.

61.  −0.2 1.4 1.2 −0.4  62.  1 __ 2 −

1 __ 2 − 1 __ 4

3 __ 4  63. 

12 9 −6

−1 3 2

−4 −3 2  64. 

2 1 3

1 2 3

3 2 1 

For the following exercises, find the solutions by computing the inverse of the matrix.

65. 0.3x − 0.1y = −10 −0.1x + 0.3y = 14

66. 0.4x − 0.2y = −0.6 −0.1x + 0.05y = 0.3

67. 4x + 3y − 3z = −4.3 5x − 4y − z = −6.1 x + z = −0.7

68. −2x − 3y + 2z = 3 −x + 2y + 4z = −5 −2y + 5z = −3

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

69. Students were asked to bring their favorite fruit to class. 90% of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were 5% more popular than bananas, what are the percentages of each individual fruit?

70. A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at \$2 and the chocolate chip cookies at \$1. They raised \$250 and sold 175 items. How many brownies and how many cookies were sold?

SOlvIng SySTemS WITh CRAmeR’S RUle

For the following exercises, find the determinant.

71.  100 0 0 0  72.  0.2 −0.6 0.7 −1.1

 73.  −1 4 3

0 2 3 0 0 −3

 74.  √ — 2 0 0

0 √ — 2 0

0 0 √ — 2 

For the following exercises, use Cramer’s Rule to solve the linear systems of equations.

75. 4x − 2y = 23 −5x − 10y = −35

76. 0.2x − 0.1y = 0 −0.3x + 0.3y = 2.5

77. −0.5x + 0.1y = 0.3 −0.25x + 0.05y = 0.15

78. x + 6y + 3z = 4 2x + y + 2z = 3 3x − 2y + z = 0

79. 4x − 3y + 5z = − 5 __ 2

7x − 9y − 3z = 3 __ 2

x − 5y − 5z = 5 __ 2

80. 3 ___ 10 x − 1 __ 5 y −

3 ___ 10 z = − 1 ___ 50

1 ___ 10 x − 1 ___ 10 y −

1 __ 2 z = − 9 ___ 50

2 __ 5 x − 1 __ 2 y −

3 __ 5 z = − 1 __ 5

979CHAPTER 11 Practice test

ChAPTeR 11 PRACTICe TeST

Is the following ordered pair a solution to the system of equations?

1. −5x − y = 12 x + 4y = 9 with ( − 3, 3)

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

2. 1 __ 2 x − 1 __ 3 y = 4

3 __ 2 x − y = 0

3. − 1 __ 2 x − 4y = 4

2x + 16y = 2

4. 5x − y = 1 −10x + 2y = − 2

5. 4x − 6y − 2z = 1 ___ 10

x − 7y + 5z = − 1 __ 4

3x + 6y − 9z = 6 __ 5

6. x + z = 20 x + y + z = 20 x + 2y + z = 10

7. 5x − 4y − 3z = 0 2x + y + 2z = 0 x − 6y − 7z = 0

8. y = x 2 + 2x − 3 y = x − 1

9. y 2 + x 2 = 25 y 2 − 2 x 2 = 1

For the following exercises, graph the following inequalities.

10. y < x 2 + 9 11. x 2 + y 2 > 4 y < x 2 + 1

For the following exercises, write the partial fraction decomposition.

12. −8x − 30 ____________ x 2 + 10x + 25

13. 13x + 2 ________ (3x + 1) 2

14. x 4 − x 3 + 2x − 1 ______________

x ( x 2 + 1) 2

For the following exercises, perform the given matrix operations.

15. 5  4 9 −2 3  + 1 __ 2 

−6 12 4 −8

 16.  1 4 −7

−2 9 5 12 0 −4

  3 −4

1 3 5 10

 17.  1 __ 2

1 __ 3 1 __ 4

1 __ 5 

−1

18. det ∣ 0 0 400 4,000 ∣ 19. det ∣ 1 __ 2 − 1 __ 2 0 − 1 __ 2 0 1 __ 2 0 1 __ 2 0 ∣ 20. If det(A) = −6, what would be the determinant if

you switched rows 1 and 3, multiplied the second row by 12, and took the inverse?

21. Rewrite the system of linear equations as an augmented matrix. 14x − 2y + 13z = 140 −2x + 3y − 6z = −1 x − 5y + 12z = 11

22. Rewrite the augmented matrix as a system of linear equations.

 1 0 3

−2 4 9 −6 1 2

∣ 12 −5 8  This OpenStax book is available for free at http://cnx.org/content/col11758/latest

980 CHAPTER 11 systems of equatioNs aNd iNequalities

For the following exercises, use Gaussian elimination to solve the systems of equations.

23. x − 6y = 4 2x − 12y = 0

24. 2x + y + z = −3 x − 2y + 3z = 6 x − y − z = 6

For the following exercises, use the inverse of a matrix to solve the systems of equations.

25. 4x − 5y = −50 −x + 2y = 80

26. 1 ___ 100 x − 3 ___ 100 y +

1 ___ 20 z = −49

3 ___ 100 x − 7 ___ 100 y −

1 ___ 100 z = 13

9 ___ 100 x − 9 ___ 100 y −

9 ___ 100 z = 99

For the following exercises, use Cramer’s Rule to solve the systems of equations.

27. 200x − 300y = 2 400x + 715y = 4

28. 0.1x + 0.1y − 0.1z = −1.2 0.1x − 0.2y + 0.4z = −1.2 0.5x − 0.3y + 0.8z = −5.9

For the following exercises, solve using a system of linear equations.

29. A factory producing cell phones has the following cost and revenue functions: C(x) = x 2 + 75x + 2,688 and R(x) = x 2 + 160x. What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

30. A small fair charges \$1.50 for students, \$1 for children, and \$2 for adults. In one day, three times as many children as adults attended. A total of 800 tickets were sold for a total revenue of \$1,050. How many of each type of ticket was sold?

13

1055

ChAPTeR OUTlIne

13.1 Sequences and Their notations 13.2 Arithmetic Sequences 13.3 geometric Sequences 13.4 Series and Their notations 13.5 Counting Principles 13.6 binomial Theorem 13.7 Probability

Figure 1 (credit: Robert S. donovan, Flickr.)

Introduction A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a villa in Saint Barthélemy? A luxury convertible? A cruise around the world?

The likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over the next 30 years or so.

This decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There are also personal reasons to consider when making the choice, and one can make many arguments for either decision. However, most lottery winners opt for the lump sum.

In this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning.

Sequences, Probability and Counting Theory

1066 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

13.1 SeCTIOn exeRCISeS

veRbAl 1. Discuss the meaning of a sequence. If a finite

sequence is defined by a formula, what is its domain? What about an infinite sequence?

2. Describe three ways that a sequence can be defined.

3. Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

4. What happens to the terms an of a sequence when there is a negative factor in the formula that is raised to a power that includes n? What is the term used to describe this phenomenon?

5. What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.

AlgebRAIC For the following exercises, write the first four terms of the sequence.

6. an = 2 n − 2 7. an = −

16 _____ n + 1 8. an = −(−5) n − 1 9. an =

2n __ n3

10. an = 2n + 1 ______ n3 11. an = 1.25 ⋅ (−4)

n − 1 12. an = −4 ⋅ (−6) n − 1 13. an =

n2 ______ 2n + 1

14. an = (−10) n + 1 15. an = −  4 ⋅ (−5)n − 1 __________ 5 

For the following exercises, write the first eight terms of the piecewise sequence.

16. an = { (−2)n − 2 if n is even (3)n − 1 if n is odd 17. an = { n 2 _

2n + 1 if n ≤ 5

n2 − 5 if n > 5

18. an = { (2n + 1)2 if n is divisible by 4 2 _ n if n is not divisible by 4 19. an = { −0.6 ⋅ 5 n − 1 if n is prime or 1

2.5 ⋅ (−2)n − 1 if n is composite

20. an = { 4(n2 − 2) if n ≤ 3 or n > 6 n2 − 2 _ 4 if 3 < n ≤ 6 For the following exercises, write an explicit formula for each sequence.

21. 4, 7, 12, 19, 28, … 22. −4, 2, − 10, 14, − 34, … 23. 1, 1, 4 __ 3 , 2, 16 ___ 5 , …

24. 0, 1 − e 1 _____ 1 + e2 ,

1 − e2 _____ 1 + e3 , 1 − e3 ______ 1 + e4

, 1 − e 4

______ 1 + e5

, … 25. 1, − 1 __ 2 , 1 __ 4 , −

1 __ 8 , 1 ___ 16 , …

For the following exercises, write the first five terms of the sequence. 26. a1 = 9, an = an − 1 + n 27. a1 = 3, an = (−3)an − 1

28. a1 = −4, an = an − 1 + 2n _ an − 1 − 1

29. a1 = −1, an = (−3)n − 1

_ an − 1 − 2

30. a1 = −30, an = (2 + an − 1)  1 __ 2  n

For the following exercises, write the first eight terms of the sequence. 31. a1 =

1 ___ 24 , a2 = 1, an = (2an − 2)(3an − 1) 32. a1 = −1, a2 = 5, an = an − 2(3 − an −1)

33. a1 = 2, a2 = 10, an = 2(an −1 + 2) __ an − 2

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SECTION 13.1 sectioN exercises 1067

For the following exercises, write a recursive formula for each sequence. 34. −2.5, − 5, − 10, − 20, − 40, … 35. −8, − 6, − 3, 1, 6, … 36. 2, 4, 12, 48, 240, …

37. 35, 38, 41, 44, 47, … 38. 15, 3, 3 __ 5 , 3 ___ 25 ,

3 ___ 125 , …

For the following exercises, evaluate the factorial. 39. 6! 40.  12 ___ 6  ! 41.

12! ___ 6! 42. 100! ____ 99!

For the following exercises, write the first four terms of the sequence. 43. an =

n! __ n2 44. an = 3 ⋅ n ! _____ 4 ⋅ n ! 45. an =

n ! _________ n2 − n − 1 46. an = 100 ⋅ n ________ n(n − 1)!

gRAPhICAl

For the following exercises, graph the first five terms of the indicated sequence

47. an = (−1)n _____ n + n 48. an = { 4 + n _ 2n if n in even 3 + n if n is odd 49. a1 = 2, an = (−an − 1 + 1)2

50. an = 1, an = an − 1 + 8 51. an = (n + 1)! _______ (n − 1)!

For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.

52.

11

15 13

7 5 (1, 5)

(2, 7) (3, 9)

(4, 11) (5, 13)

3

0 1 2 3 4 5 6 7 n

an

9

53.

8 7 6 5 4 3 2 1

0 1

(4, 4)

(3, 2) (2, 1) (1, 0.5)

(5, 8)

2 3 4 5 6 n

an

7

54.

(1, 12) (2, 9)

(3, 6) (4, 3)

(5, 0)

15 18

9 6 3

0 1 2 3 4 5 6 7 n

an

12

For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph. 55.

(5, 21)

(4, 13)

(3, 9) (2, 7)

(1, 6)

20 22

16

12

8

4

0 1 2 3 4 5 n

an 56.

(3, 4)

(2, 8)

(1, 16)

(4, 2) (5, 1)

16

12

8

4

0 1 2 3 4 5 n

an

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1068 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

TeChnOlOgy Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

• On the home screen, key in the value for the initial term a1 and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes

[2ND] ANS for the previous term an − 1. Press [ENTER]. • Continue pressing [ENTER] to calculate the values for each successive term.

For the following exercises, use the steps above to find the indicated term or terms for the sequence. 57. Find the first five terms of the sequence a1 =

87 ___ 111 , an =

4 __ 3 an − 1 + 12 ___ 37 . Use the >Frac feature to give

fractional results.

58. Find the 15th term of the sequence a1 = 625, an = 0.8an − 1 + 18.

59. Find the first five terms of the sequence a1 = 2, an = 2

[(an − 1) − 1] + 1. 60. Find the first ten terms of the sequence

a1 = 8, an = (an − 1 + 1)! _

an − 1! .

61. Find the tenth term of the sequence a1 = 2, an = nan − 1

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

• In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(” from the dropdown list. Press [ENTER]. • In the line headed “Expr:” type in the explicit formula, using the [X,T, θ, n] button for n • In the line headed “Variable:” type in the variable used on the previous step. • In the line headed “start:” key in the value of n that begins the sequence. • In the line headed “end:” key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press

[ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(” from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas. See the instructions above

for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through

the list of terms.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

62. List the first five terms of the sequence. an = −

28 ___ 9 n + 5 _ 3

63. List the first six terms of the sequence.

an = n3 − 3.5n2 + 4.1n − 1.5 ___________________ 2.4n

64. List the first five terms of the sequence. an =

15n ⋅ (−2)n − 1 ____________ 47 65. List the first four terms of the sequence.

an = 5.7 n + 0.275(n − 1)!

66. List the first six terms of the sequence an = n! __ n .

exTenSIOnS 67. Consider the sequence defined by an = −6 − 8n. Is

an = −421 a term in the sequence? Verify the result. 68. What term in the sequence an =

n2 + 4n + 4 __________ 2(n + 2) has the value 41? Verify the result.

69. Find a recursive formula for the sequence 1, 0, −1, −1, 0, 1, 1, 0, −1, −1, 0, 1, 1, … (Hint: find a pattern for an based on the first two terms.)

70. Calculate the first eight terms of the sequences

an = (n + 2)! _______ (n − 1)! and bn = n

3 + 3n2 + 2n, and then make a conjecture about the relationship between these two sequences.

71. Prove the conjecture made in the preceding exercise.

1076 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

13.2 SeCTIOn exeRCISeS

veRbAl

1. What is an arithmetic sequence? 2. How is the common difference of an arithmetic sequence found?

3. How do we determine whether a sequence is arithmetic?

4. What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence?

5. Describe how linear functions and arithmetic sequences are similar. How are they different?

AlgebRAIC

For the following exercises, find the common difference for the arithmetic sequence provided. 6. {5, 11, 17, 23, 29, … } 7.  0, 1 __ 2 , 1,

3 __ 2 , 2, … 

For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.

8. {11.4, 9.3, 7.2, 5.1, 3, … } 9. {4, 16, 64, 256, 1024, … }

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.

10. a1 = −25, d = −9 11. a1 = 0, d = 2 __ 3

For the following exercises, write the first five terms of the arithmetic series given two terms. 12. a1 = 17, a7 = −31 13. a13 = −60, a33 = −160

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.

14. First term is 3, common difference is 4, find the 5th term.

15. First term is 4, common difference is 5, find the 4th term.

16. First term is 5, common difference is 6, find the 8th term.

17. First term is 6, common difference is 7, find the 6th term.

18. First term is 7, common difference is 8, find the 7th term.

For the following exercises, find the first term given two terms from an arithmetic sequence. 19. Find the first term or a1 of an arithmetic sequence if

a6 = 12 and a14 = 28. 20. Find the first term or a1 of an arithmetic sequence if

a7 = 21 and a15 = 42. 21. Find the first term or a1 of an arithmetic sequence if

a8 = 40 and a23 = 115. 22. Find the first term or a1 of an arithmetic sequence if

a9 = 54 and a17 = 102. 23. Find the first term or a1 of an arithmetic sequence if

a11 = 11 and a21 = 16.

For the following exercises, find the specified term given two terms from an arithmetic sequence. 24. a1 = 33 and a7 = −15. Find a4. 25. a3 = −17.1 and a10 = −15.7. Find a21.

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 26. a1 = 39; an = an − 1 −3 27. a1 = −19; an = an − 1 −1.4

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SECTION 13.2 sectioN exercises 1077

For the following exercises, write a recursive formula for each arithmetic sequence. 28. a = {40, 60, 80, … } 29. a = {17, 26, 35, … } 30. a = {−1, 2, 5, … }

31. a = {12, 17, 22, … } 32. a = {−15, −7, 1, … } 33. a = {8.9, 10.3, 11.7, … }

34. a = {−0.52, −1.02, −1.52, … } 35. a =  1 __ 5 , 9 ___ 20 ,

7 ___ 10

, …  36. a =  − 1 __ 2 , − 5 __ 4 , −2, … 

37. a =  1 __ 6 , − 11 ___ 12 , −2, … 

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 38. a = {7, 4, 1, … }; Find the 17th term. 39. a = {4, 11, 18, … }; Find the 14th term. 40. a = {2, 6, 10, … }; Find the 12th term.

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 41. a = 24 − 4n 42. a = 1 __ 2 n −

1 __ 2

For the following exercises, write an explicit formula for each arithmetic sequence. 43. a = {3, 5, 7, … } 44. a = {32, 24, 16, … } 45. a = {−5, 95, 195, … } 46. a = {−17, −217, −417, … } 47. a = {1.8, 3.6, 5.4, … } 48. a = {−18.1, −16.2, −14.3, … }

49. a = {15.8, 18.5, 21.2, … } 50. a =  1 __ 3 , − 4 __ 3 , −3, …  51. a =  0,

1 __ 3 , 2 __ 3 , … 

52. a =  −5, − 10 ___ 3 , − 5 __ 3 , … 

For the following exercises, find the number of terms in the given finite arithmetic sequence. 53. a = {3, −4, −11, … , −60} 54. a = {1.2, 1.4, 1.6, … , 3.8} 55. a =  1 __ 2 , 2,

7 __ 2 , … , 8  gRAPhICAl

For the following exercises, determine whether the graph shown represents an arithmetic sequence.

56.

0.50 −0.5

−0.5

−1.5

−3

−4

−5

−2.5

−3.5

−4.5

−5.5

−1

−2

0.5

1.5

2.5

3.5

4.5

5.5

n 1.5 2.5 3.5

(3, 0)

(2, −2)

(1, −4)

(4, 2)

(5, 4)

4.5 5.52 3 4 51

1

2

3

4

5

an 57.

0.5

−0.5 −0.5

0 0.5

(1, 1.5)

(2, 2.25)

(3, 3.375)

(4, 5.0625)

(5, 7.5938)

1.5 2.5 3.5 4.5 5.52 3 4 51

1.5 2

2.5

3.5

4.5

5.5

6.5

7.5

8.5

7

8

6

5

4

3

1

n

an

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1078 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. 58. a1 = 0, d = 4 59. a1 = 9; an = an − 1 − 10 60. an = −12 + 5n

TeChnOlOgy

For the following exercises, follow the steps to work with the arithmetic sequence an = 3n − 2 using a graphing

calculator: • Press [MODE]

› Select [SEQ] in the fourth line › Select [DOT] in the fifth line › Press [ENTER]

• Press [Y=] › nMin is the first counting number for the sequence. Set nMin = 1 › u(n) is the pattern for the sequence. Set u(n) = 3n − 2 › u(nMin) is the first number in the sequence. Set u(nMin) = 1

• Press [2ND] then [WINDOW] to go to TBLSET › Set TblStart = 1 › Set ΔTbl = 1 › Set Indpnt: Auto and Depend: Auto

• Press [2ND] then [GRAPH] to go to the [TABLE]

61. What are the first seven terms shown in the column with the heading u(n)?

62. Use the scroll-down arrow to scroll to n = 50. What value is given for u(n)?

63. Press [WINDOW]. Set nMin = 1, nMax = 5, xMin = 0, xMax = 6, yMin = −1, and yMax = 14. Then press [GRAPH]. Graph the sequence as it appears on the graphing calculator.

For the following exercises, follow the steps given above to work with the arithmetic sequence an = 1 __ 2 n + 5 using a graphing calculator.

64. What are the first seven terms shown in the column with the heading u(n) in the [TABLE] feature?

65. Graph the sequence as it appears on the graphing calculator. Be sure to adjust the [WINDOW] settings as needed.

exTenSIOnS 66. Give two examples of arithmetic sequences whose

4th terms are 9. 67. Give two examples of arithmetic sequences whose

10th terms are 206.

68. Find the 5th term of the arithmetic sequence {9b, 5b, b, … }.

69. Find the 11th term of the arithmetic sequence {3a − 2b, a + 2b, −a + 6b, … }.

70. At which term does the sequence {5.4, 14.5, 23.6, …} exceed 151?

71. At which term does the sequence  17 _ 3 , 31 _ 6

, 14 _ 3

,…  begin to have negative values?

72. For which terms does the finite arithmetic sequence

 5 _ 2 , 19 ___ 8 ,

9 _ 4 , … , 1 _ 8

 have integer values? 73. Write an arithmetic sequence using a recursive

formula. Show the first 4 terms, and then find the 31st term.

74. Write an arithmetic sequence using an explicit formula. Show the first 4 terms, and then find the 28th term.

SECTION 13.3 sectioN exercises 1085

13.3 SeCTIOn exeRCISeS

veRbAl

1. What is a geometric sequence? 2. How is the common ratio of a geometric sequence found?

3. What is the procedure for determining whether a sequence is geometric?

4. What is the difference between an arithmetic sequence and a geometric sequence?

5. Describe how exponential functions and geometric sequences are similar. How are they different?

AlgebRAIC For the following exercises, find the common ratio for the geometric sequence.

6. 1, 3, 9, 27, 81, … 7. −0.125, 0.25, −0.5, 1, −2, … 8. −2, − 1 ___ 2 , − 1 ___ 8 , −

1 ___ 32 , − 1 ___ 128 , …

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

9. −6, −12, −24, −48, −96, … 10. 5, 5.2, 5.4, 5.6, 5.8, … 11. −1, 1 __ 2 , − 1 ___ 4 ,

1 __ 8 , − 1 ___ 16 , …

12. 6, 8, 11, 15, 20, … 13. 0.8, 4, 20, 100, 500, …

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.

14. a1 = 8, r = 0.3 15. a1 = 5, r = 1 __ 5

For the following exercises, write the first five terms of the geometric sequence, given any two terms. 16. a7 = 64, a10 = 512 17. a6 = 25, a8 = 6.25

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio. 18. The first term is 2, and the common ratio is 3. Find

the 5th term. 19. The first term is 16 and the common ratio is − 1 ___ 3 .

Find the 4th term.

For the following exercises, find the specified term for the geometric sequence, given the first four terms.

20. an = {−1, 2, −4, 8, …}. Find a12. 21. an =  −2, 2 __ 3 , − 2 __ 9 ,

2 ___ 27 , ….  Find a 7 .

For the following exercises, write the first five terms of the geometric sequence.

22. a1 = −486, an = − 1 ___ 3 an − 1 23. a1 = 7, an = 0.2an − 1

For the following exercises, write a recursive formula for each geometric sequence. 24. an = {−1, 5, −25, 125, …} 25. an = {−32, −16, −8, −4, …}

26. an = {14, 56, 224, 896, …} 27. an = {10, −3, 0.9, −0.27, …}

28. an = {0.61, 1.83, 5.49, 16.47, …} 29. an =  3 __ 5 , 1 ___ 10 ,

1 ___ 60 , 1 ___ 360 , … 

30. an =  −2, 4 __ 3 , − 8 __ 9 ,

16 ___ 27 , …  31. an =  1 ___ 512 , −

1 ___ 128 , 1 ___ 32 , −

1 ___ 8 , … 

For the following exercises, write the first five terms of the geometric sequence. 32. an = −4 ⋅ 5

n − 1 33. an = 12 ⋅  − 1 ___ 2 

n − 1

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1086 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

For the following exercises, write an explicit formula for each geometric sequence. 34. an = {−2, −4, −8, −16, …} 35. an = {1, 3, 9, 27, …} 36. an = {−4, −12, −36, −108, …} 37. an = {0.8, −4, 20, −100, …} 38. an = {−1.25, −5, −20, −80, …} 39. an =  −1, − 4 __ 5 , −

16 ___ 25 , − 64 ___ 125 , … 

40. an =  2, 1 __ 3 , 1 ___ 18 ,

1 ___ 108 , …  41. an =  3, −1, 1 __ 3 , −

1 __ 9 , …  For the following exercises, find the specified term for the geometric sequence given.

42. Let a1 = 4, an = −3an − 1. Find a8. 43. Let an = −  − 1 __ 3  n − 1

. Find a12.

For the following exercises, find the number of terms in the given finite geometric sequence. 44. an = {−1, 3, −9, … , 2187} 45. an =  2, 1, 1 __ 2 , … ,

1 ____ 1024  gRAPhICAl For the following exercises, determine whether the graph shown represents a geometric sequence.

46.

5 6

2

0 0.5 1 2 2.5 3.5 n

an

4 3

1

−1 −2 −3 −4

−0.5 1.5 3

(3, 1)

(2, −1)

(1, −3)

(4, 3)

(5, 5)

4 4.5 5 5.5

47.

5.5

0 0.5 1 2 2.5 3.5 n

an 6

5 4.5

4 3.5

3 2.5

2 1.5

1 0.5

−0.5 −1

−0.5 1.5 3

(2, 0.25)

(1, −0.5) 4 4.5 5 5.5

(3, 1.375)

(4, 3.0625)

(5, 5.5938)

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

48. a1 = 1, r = 1 __ 2 49. a1 = 3, an = 2an − 1 50. an = 27 ⋅ 0.3

n − 1

exTenSIOnS 51. Use recursive formulas to give two examples of

geometric sequences whose 3rd terms are 200. 52. Use explicit formulas to give two examples of

geometric sequences whose 7th terms are 1024.

53. Find the 5th term of the geometric sequence {b, 4b, 16b, …}.

54. Find the 7th term of the geometric sequence {64a(−b), 32a(−3b), 16a(−9b), …}.

55. At which term does the sequence {10, 12, 14.4, 17.28, …} exceed 100?

56. At which term does the sequence

 1 ____ 2187 , 1 ___ 729 ,

1 ___ 243 , 1 ___ 81 …  begin to have integer values?

57. For which term does the geometric sequence

an = −36  2 __ 3  n − 1

first have a non-integer value?

58. Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10th term.

59. Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8th term.

60. Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.

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SECTION 13.4 sectioN exercises 1097

13.4 SeCTIOn exeRCISeS

veRbAl

1. What is an nth partial sum? 2. What is the difference between an arithmetic sequence and an arithmetic series?

3. What is a geometric series? 4. How is finding the sum of an infinite geometric series different from finding the nth partial sum?

5. What is an annuity?

AlgebRAIC

For the following exercises, express each description of a sum using summation notation. 6. The sum of terms m2 + 3m from m = 1 to m = 5 7. The sum from of n = 0 to n = 4 of 5n

8. The sum of 6k − 5 from k = −2 to k = 1 9. The sum that results from adding the number 4 five times

For the following exercises, express each arithmetic sum using summation notation. 10. 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 11. 10 + 18 + 26 + … + 162

12. 1 __ 2 + 1 + 3 ___ 2 + 2 + … + 4

For the following exercises, use the formula for the sum of the first n terms of each arithmetic sequence.

13. 3 __ 2 + 2 + 5 __ 2 + 3 +

7 __ 2 14. 19 + 25 + 31 + … + 73 15. 3.2 + 3.4 + 3.6 + … + 5.6

For the following exercises, express each geometric sum using summation notation. 16. 1 + 3 + 9 + 27 + 81 + 243 + 729 + 2187 17. 8 + 4 + 2 + … + 0.125

18. − 1 __ 6 + 1 ___ 12 −

1 ___ 24 + … + 1 ___ 768

For the following exercises, use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum.

19. 9 + 3 + 1 + 1 _ 3

+ 1 _ 9

20. ∑ n = 1

9

5 ⋅ 2n − 1 21. ∑ a = 1

11

64 ⋅ 0.2a − 1

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

22. 12 + 18 + 24 + 30 + … 23. 2 + 1.6 + 1.28 + 1.024 + … 24. ∑ m = 1

4m − 1

25. ∑ k = 1

−  − 1 __ 2  k − 1

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1098 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

gRAPhICAl

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of \$50. Each month thereafter he increased the previous deposit amount by \$20.

26. Graph the arithmetic sequence showing one year of Javier’s deposits.

27. Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits.

For the following exercises, use the geometric series ∑ k = 1

 1 __ 2  k .

28. Graph the first 7 partial sums of the series. 29. What number does Sn seem to be approaching in the graph? Find the sum to explain why this makes sense.

nUmeRIC For the following exercises, find the indicated sum.

30. ∑ a = 1

14

a 31. ∑ n = 1

6

n(n − 2) 32. ∑ k = 1

17

k2 33. ∑ k = 1

7

2k

For the following exercises, use the formula for the sum of the first n terms of an arithmetic series to find the sum.

34. −1.7 + −0.4 + 0.9 + 2.2 + 3.5 + 4.8 35. 6 + 15 ___ 2 + 9 + 21 ___ 2 + 12 +

27 ___ 2 + 15

36. −1 + 3 + 7 + … + 31 37. ∑ k = 1

11

 k __ 2 − 1 __ 2 

For the following exercises, use the formula for the sum of the first n terms of a geometric series to find the partial sum.

38. S6 for the series −2 − 10 − 50 − 250 … 39. S7 for the series 0.4 − 2 + 10 − 50 …

40. ∑ k = 1

9

2k − 1 41. ∑ n = 1

10

−2 ⋅  1 __ 2  n − 1

For the following exercises, find the sum of the infinite geometric series.

42. 4 + 2 + 1 + 1 __ 2 … 43. −1 − 1 __ 4 −

1 ___ 16 − 1 ___ 64 … 44. ∑ ∞

k = 1

3 ⋅  1 __ 4  k − 1

45. ∑ n = 1

4.6 ⋅ 0.5n − 1

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

46. Deposit amount: \$50; total deposits: 60; interest rate: 5%, compounded monthly

47. Deposit amount: \$150; total deposits: 24; interest rate: 3%, compounded monthly

48. Deposit amount: \$450; total deposits: 60; interest rate: 4.5%, compounded quarterly

49. Deposit amount: \$100; total deposits: 120; interest rate: 10%, compounded semi-annually

exTenSIOnS 50. The sum of terms 50 − k 2 from k = x through 7 is

115. What is x? 51. Write an explicit formula for ak such that

∑ k = 0

6

ak = 189 . Assume this is an arithmetic series.

52. Find the smallest value of n such that

∑ k = 1

n

(3k − 5) > 100 .

53. How many terms must be added before the series −1 − 3 − 5 − 7…. has a sum less than −75?

SECTION 13.4 sectioN exercises 1099

54. Write 0. _

65 as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert 0.65 to a fraction.

55. The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?

56. To get the best loan rates available, the Riches want to save enough money to place 20% down on a \$160,000 home. They plan to make monthly deposits of \$125 in an investment account that offers 8.5% annual interest compounded semi- annually. Will the Riches have enough for a 20% down payment after five years of saving? How much money will they have saved?

57. Karl has two years to save \$10,000 to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?

ReAl-WORld APPlICATIOnS 58. Keisha devised a week-long study plan to prepare for

finals. On the first day, she plans to study for 1 hour, and each successive day she will increase her study time by 30 minutes. How many hours will Keisha have studied after one week?

59. A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?

60. A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day?

61. A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels 3 __ 4 the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?

62. Rachael deposits \$1,500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?

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SECTION 13.6 sectioN exercises 1115

13.6 SeCTIOn exeRCISeS

veRbAl

1. What is a binomial coefficient, and how it is calculated?

2. What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?

3. What is the Binomial Theorem and what is its use? 4. When is it an advantage to use the Binomial Theorem? Explain.

AlgebRAIC

For the following exercises, evaluate the binomial coefficient.

5.  6 _ 2  6.  5 _ 3

 7.  7 _ 4  8.  9 _ 7  9. 

10 _ 9

 10.  25 _ 11  11. 

17 _ 6

 12.  200 _ 199 

For the following exercises, use the Binomial Theorem to expand each binomial.

13. (4a − b)3 14. (5a + 2)3 15. (3a + 2b)3 16. (2x + 3y)4 17. (4x + 2y)5

18. (3x − 2y)4 19. (4x − 3y)5 20.  1 _ x + 3y  5 21. (x−1 + 2y −1)4 22. ( √— x − √— y )5

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

23. (a + b)17 24. (x − 1)18 25. (a − 2b)15 26. (x − 2y)8 27. (3a + b)20

28. (2a + 4b)7 29. (x3 − √— y )8

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

30. The fourth term of (2x − 3y)4 31. The fourth term of (3x − 2y)5

32. The third term of (6x − 3y)7 33. The eighth term of (7 + 5y)14

34. The seventh term of (a + b)11 35. The fifth term of (x − y)7

36. The tenth term of (x − 1)12 37. The ninth term of (a − 3b 2)11

38. The fourth term of  x 3 − 1 __ 2  10

39. The eighth term of  y _ 2 + 2 _ x 

9

gRAPhICAl

For the following exercises, use the Binomial Theorem to expand the binomial f (x) = (x + 3)4. Then find and graph each indicated sum on one set of axes.

40. Find and graph f1(x), such that f1(x) is the first term of the expansion.

41. Find and graph f2(x), such that f2(x) is the sum of the first two terms of the expansion.

42. Find and graph f3(x), such that f3(x) is the sum of the first three terms of the expansion.

43. Find and graph f4(x), such that f4(x) is the sum of the first four terms of the expansion.

44. Find and graph f5(x), such that f5(x) is the sum of the first five terms of the expansion.

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1116 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

exTenSIOnS

45. In the expansion of (5x + 3y)n, each term has the form  n _ k  a

n − kbk, where k successively takes on

the value 0, 1, 2, …, n. If  n _ k  =  7 _ 2

 , what is the corresponding term?

46. In the expansion of (a + b)n, the coefficient of a n − kbk is the same as the coefficient of which other term?

47. Consider the expansion of (x + b)40. What is the exponent of b in the kth term?

48. Find  n _ k − 1  +  n _ k

 and write the answer as a binomial coefficient in the form  n _ k  . Prove it. Hint: Use the fact that, for any integer p, such that p ≥ 1, p! = p(p − 1)!.

49. Which expression cannot be expanded using the Binomial Theorem? Explain. a. (x 2 − 2x + 1) b. ( √— a + 4 √— a − 5)8 c. (x 3 + 2y 2 − z)5

d. (3x 2 − √— 2y 3 )12

SECTION 13.7 Probability 1117

leARnIng ObjeCTIveS

In this section, you will:

• Construct probability models.

• Compute probabilities of equally likely outcomes.

• Compute probabilities of the union of two events.

• Use the complement rule to find probabilities.

• Compute probability using counting theory.

13.7 PRObAbIlITy

Figure 1 An example of a “spaghetti model,” which can be used to predict possible paths of a tropical storm.[34]

Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in Figure 1. They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.

Constructing Probability models Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is {1, 2, 3, 4, 5, 6}. An event is any subset of a sample space.

The likelihood of an event is known as probability. The probability of an event p is a number that always satisfies 0 ≤ p ≤ 1, where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like Table 1.

Outcome Probability Winning the raffle 1%

Losing the raffle 99% Table 1

The sum of the probabilities listed in a probability model must equal 1, or 100%.

34 The figure is for illustrative purposes only and does not model any particular storm.

CHAPTER 13 sequeNces, Probability aNd couNtiNg theory1118

How To… Given a probability event where each event is equally likely, construct a probability model. 1. Identify every outcome. 2. Determine the total number of possible outcomes. 3. Compare each outcome to the total number of possible outcomes.

Example 1 Constructing a Probability Model

Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.

Solution Begin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be rolled: 1, 2, 3, 4, 5, and 6. There are six possible outcomes that make up the sample space.

Assign probabilities to each outcome in the sample space by determining a ratio of the outcome to the number of possible outcomes. There is one of each of the six numbers on the cube, and there is no reason to think that any particular face is more likely to show up than any other one, so the probability of rolling any number is 1 __ 6 .

Outcome Roll of 1 Roll of 2 Roll of 3 Roll of 4 Roll of 5 Roll of 6

Probability 1 __ 6 1 __ 6

1 __ 6 1 __ 6

1 __ 6 1 __ 6

Table 2

Q & A… Do probabilities always have to be expressed as fractions? No. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a number between 0 and 1, inclusive of 0 and 1.

Try It #1 Construct a probability model for tossing a fair coin.

Computing Probabilities of equally likely Outcomes Let S be a sample space for an experiment. When investigating probability, an event is any subset of S. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in S. Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the

event and 6 possible outcomes in S, so the probability of the event is 4 __ 6 = 2 __ 3 .

computing the probability of an event with equally likely outcomes The probability of an event E in an experiment with sample space S with equally likely outcomes is given by

P(E) = number of elements in E ____________________ number of elements in S = n(E) ____ n(S)

E is a subset of S, so it is always true that 0 ≤ P(E) ≤ 1.

Example 2 Computing the Probability of an Event with Equally Likely Outcomes

A six-sided number cube is rolled. Find the probability of rolling an odd number.

Solution The event “rolling an odd number” contains three outcomes. There are 6 equally likely outcomes in the sample space. Divide to find the probability of the event.

P(E) = 3 __ 6 = 1 __ 2

SECTION 13.7 Probability 1119

Try It #2

A six-sided number cube is rolled. Find the probability of rolling a number greater than 2.

Computing the Probability of the Union of Two events We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events E and F, written E ∪ F, is the event that occurs if either or both events occur.

P(E ∪ F) = P(E) + P(F) − P(E ∩ F)

Suppose the spinner in Figure 2 is spun. We want to find the probability of spinning orange or spinning a b.

a

a

b

b

cd

Figure 2

There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is 3 __ 6 = 1 __ 2 . There

are a total of 6 sections, and 2 of them have a b. So the probability of spinning a b is 2 __ 6 = 1 __ 3 . If we added these two

probabilities, we would be counting the sector that is both orange and a b twice. To find the probability of spinning an orange or a b, we need to subtract the probability that the sector is both orange and has a b.

1 __ 2 + 1 __ 3 −

1 __ 6 = 2 __ 3

The probability of spinning orange or a b is 2 __ 3 .

probability of the union of two events The probability of the union of two events E and F (written E ∪ F) equals the sum of the probability of E and the probability of F minus the probability of E and F occurring together (which is called the intersection of E and F and is written as E ∩ F). P(E ∪ F) = P(E) + P(F) − P(E ∩ F)

Example 3 Computing the Probability of the Union of Two Events

A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.

Solution A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is 1 __ 4 . There are four 7s in a standard deck, and there are a total of 52 cards. So the probability of drawing a 7 is 1 ___ 13 .

The only card in the deck that is both a heart and a 7 is the 7 of hearts, so the probability of drawing both a heart and

a 7 is 1 ___ 52 . Substitute P(H) = 1 __ 4 , P(7) =

1 ___ 13 , and P(H ∩ 7) = 1 ___ 52 into the formula.

P(E ∪ F) = P(E) + P(F) − P(E ∩ F)

= 1 __ 4 + 1 ___ 13 −

1 ___ 52

= 4 ___ 13

The probability of drawing a heart or a 7 is 4 ___ 13 .

CHAPTER 13 sequeNces, Probability aNd couNtiNg theory1120

Try It #3

A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

Computing the Probability of mutually exclusive events Suppose the spinner in Figure 2 is spun again, but this time we are interested in the probability of spinning an orange or a d. There are no sectors that are both orange and contain a d, so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is

P(E ∪ F) = P(E) + P(F)

Notice that with mutually exclusive events, the intersection of E and F is the empty set. The probability of spinning

an orange is 3 __ 6 = 1 __ 2 and the probability of spinning a d is

1 __ 6 . We can find the probability of spinning an orange or a

d simply by adding the two probabilities.

P(E ∪ F) = P(E) + P(F)

= 1 __ 2 + 1 __ 6

= 2 __ 3

The probability of spinning an orange or a d is 2 __ 3 .

probability of the union of mutually exclusive events The probability of the union of two mutually exclusive events E and F is given by

P(E ∪ F) = P(E) + P(F)

How To… Given a set of events, compute the probability of the union of mutually exclusive events.

1. Determine the total number of outcomes for the first event. 2. Find the probability of the first event. 3. Determine the total number of outcomes for the second event. 4. Find the probability of the second event. 5. Add the probabilities.

Example 4 Computing the Probability of the Union of Mutually Exclusive Events

A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.

Solution The events “drawing a heart” and “drawing a spade” are mutually exclusive because they cannot occur at the same time. The probability of drawing a heart is 1 __ 4 , and the probability of drawing a spade is also

1 __ 4 , so the

probability of drawing a heart or a spade is

1 __ 4 + 1 __ 4 =

1 __ 2

Try It #4

A card is drawn from a standard deck. Find the probability of drawing an ace or a king. Using the

SECTION 13.7 Probability 1121

Complement Rule to Compute Probabilities We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event E, denoted E′, is the set of outcomes in the sample space that are not in E. For example, suppose we are interested in the probability that a horse will lose a race. If event W is the horse winning the race, then the complement of event W is the horse losing the race.

To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1. P(E′) = 1 − P(E)

The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the

probability of the horse winning the race is 1 __ 9 , the probability of the horse losing the race is simply

1 − 1 __ 9 = 8 __ 9

the complement rule The probability that the complement of an event will occur is given by

P(E′) = 1 − P(E)

Example 5 Using the Complement Rule to Calculate Probabilities

Two six-sided number cubes are rolled.

a. Find the probability that the sum of the numbers rolled is less than or equal to 3.

b. Find the probability that the sum of the numbers rolled is greater than 3.

Solution The first step is to identify the sample space, which consists of all the possible outcomes. There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are 6 × 6, or 36 total possible outcomes. So, for example, 1-1 represents a 1 rolled on each number cube.

1-1 1-2 1-3 1-4 1-5 1-6 2-1 2-2 2-3 2-4 2-5 2-6 3-1 3-2 3-3 3-4 3-5 3-6 4-1 4-2 4-3 4-4 4-5 4-6 5-1 5-2 5-3 5-4 5-5 5-6 6-1 6-2 6-3 6-4 6-5 6-6

Table 3

a. We need to count the number of ways to roll a sum of 3 or less. These would include the following outcomes: 1-1, 1-2, and 2-1. So there are only three ways to roll a sum of 3 or less. The probability is

3 ___ 36 = 1 ___ 12

b. Rather than listing all the possibilities, we can use the Complement Rule. Because we have already found the probability of the complement of this event, we can simply subtract that probability from 1 to find the probability that the sum of the numbers rolled is greater than 3.

P(E′) = 1 − P(E)

= 1 − 1 ___ 12

= 11 ___ 12

Try It #5

Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.

CHAPTER 13 sequeNces, Probability aNd couNtiNg theory1122

Computing Probability Using Counting Theory Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.

Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are C(5, 2) ways to select 2 phones that are not defective. There are 8 phones, so there are C(8, 2) ways to select 2 phones. The probability of selecting 2 phones that are not defective is:

ways to select 2 phones that are not defective

____ ways to select 2 phones

= C(5, 2)

_ C(8, 2)

= 10 ___ 28

= 5 ___ 14

Example 6 Computing Probability Using Counting Theory

A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears.

a. Find the probability that only bears are chosen.

b. Find the probability that 2 bears and 3 dogs are chosen.

c. Find the probability that at least 2 dogs are chosen.

Solution

a. We need to count the number of ways to choose only bears and the total number of possible ways to select 5 toys. There are 6 bears, so there are C(6, 5) ways to choose 5 bears. There are 14 toys, so there are C(14, 5) ways to choose any 5 toys.

C(6, 5) _______ C(14, 5) = 6 _____ 2,002 =

3 _____ 1,001

b. We need to count the number of ways to choose 2 bears and 3 dogs and the total number of possible ways to select 5 toys. There are 6 bears, so there are C(6, 2) ways to choose 2 bears. There are 5 dogs, so there are C(5, 3) ways to choose 3 dogs. Since we are choosing both bears and dogs at the same time, we will use the Multiplication Principle. There are C(6, 2) ⋅ C(5, 3) ways to choose 2 bears and 3 dogs. We can use this result to find the probability.

C(6, 2)C(5, 3) ____________ C(14, 5) = 15 ⋅ 10 ______ 2,002 =

75 _____ 1,001

c. It is often easiest to solve “at least” problems using the Complement Rule. We will begin by finding the probability that fewer than 2 dogs are chosen. If less than 2 dogs are chosen, then either no dogs could be chosen, or 1 dog could be chosen.

When no dogs are chosen, all 5 toys come from the 9 toys that are not dogs. There are C(9, 5) ways to choose toys from the 9 toys that are not dogs. Since there are 14 toys, there are C(14, 5) ways to choose the 5 toys from all of the toys.

C(9, 5) _______ C(14,5) = 63 _____ 1,001

If there is 1 dog chosen, then 4 toys must come from the 9 toys that are not dogs, and 1 must come from the 5 dogs. Since we are choosing both dogs and other toys at the same time, we will use the Multiplication Principle. There are C(5, 1) ⋅ C(9, 4) ways to choose 1 dog and 1 other toy.

C(5, 1)C(9, 4) ____________ C(14, 5) = 5 ⋅ 126 ______ 2,002 =

315 _____ 1,001

SECTION 13.7 Probability 1123

Because these events would not occur together and are therefore mutually exclusive, we add the probabilities to find the probability that fewer than 2 dogs are chosen.

63 _____ 1,001 + 315 _____ 1,001 =

378 _____ 1,001

We then subtract that probability from 1 to find the probability that at least 2 dogs are chosen.

1 − 378 _____ 1,001 = 623 _____ 1,001

Try It #6

A child randomly selects 3 gumballs from a container holding 4 purple gumballs, 8 yellow gumballs, and 2 green gumballs.

a. Find the probability that all 3 gumballs selected are purple.

b. Find the probability that no yellow gumballs are selected.

c. Find the probability that at least 1 yellow gumball is selected.

Access these online resources for additional instruction and practice with probability.

• Introduction to Probability (http://openstaxcollege.org/l/introprob)

• determining Probability (http://openstaxcollege.org/l/determineprob)

1126 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

ChAPTeR 13 RevIeW

Key Terms Addition Principle if one event can occur in m ways and a second event with no common outcomes can occur in n ways,

then the first or second event can occur in m + n ways

annuity an investment in which the purchaser makes a sequence of periodic, equal payments

arithmetic sequence a sequence in which the difference between any two consecutive terms is a constant

arithmetic series the sum of the terms in an arithmetic sequence

binomial coefficient the number of ways to choose r objects from n objects where order does not matter; equivalent to C(n, r), denoted  n _ r 

binomial expansion the result of expanding (x + y)n by multiplying

Binomial Theorem a formula that can be used to expand any binomial

combination a selection of objects in which order does not matter

common difference the difference between any two consecutive terms in an arithmetic sequence

common ratio the ratio between any two consecutive terms in a geometric sequence

complement of an event the set of outcomes in the sample space that are not in the event E

diverge a series is said to diverge if the sum is not a real number

event any subset of a sample space

experiment an activity with an observable result

explicit formula a formula that defines each term of a sequence in terms of its position in the sequence

finite sequence a function whose domain consists of a finite subset of the positive integers {1, 2, … n} for some positive integer n

Fundamental Counting Principle if one event can occur in m ways and a second event can occur in n ways after the first event has occurred, then the two events can occur in m × n ways; also known as the Multiplication Principle

geometric sequence a sequence in which the ratio of a term to a previous term is a constant

geometric series the sum of the terms in a geometric sequence

index of summation in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation

infinite sequence a function whose domain is the set of positive integers

infinite series the sum of the terms in an infinite sequence

lower limit of summation the number used in the explicit formula to find the first term in a series

Multiplication Principle if one event can occur in m ways and a second event can occur in n ways after the first event has occurred, then the two events can occur in m × n ways; also known as the Fundamental Counting Principle

mutually exclusive events events that have no outcomes in common

n factorial the product of all the positive integers from 1 to n

nth partial sum the sum of the first n terms of a sequence

nth term of a sequence a formula for the general term of a sequence

outcomes the possible results of an experiment

permutation a selection of objects in which order matters

probability a number from 0 to 1 indicating the likelihood of an event

probability model a mathematical description of an experiment listing all possible outcomes and their associated probabilities

CHAPTER 13 review 1127

recursive formula a formula that defines each term of a sequence using previous term(s )

sample space the set of all possible outcomes of an experiment

sequence a function whose domain is a subset of the positive integers

series the sum of the terms in a sequence

summation notation a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series

term a number in a sequence

union of two events the event that occurs if either or both events occur

upper limit of summation the number used in the explicit formula to find the last term in a series

Key equations 0! = 1 Formula for a factorial 1! = 1 n!= n(n − 1)(n − 2) ⋯ (2)(1), for n ≥ 2

recursive formula for nth term of an arithmetic sequence an = an −1 + d; n ≥ 2

explicit formula for nth term of an arithmetic sequence an = a1 + d(n − 1)

recursive formula for nth term of a geometric sequence an = ran − 1, n ≥ 2

explicit formula for nth term of a geometric sequence an = a1r n −1

sum of the first n terms of an arithmetic series Sn = n(a1+ an) _

2

sum of the first n terms of a geometric series Sn = a1(1 − r

n) _________ 1 − r , r ≠1

sum of an infinite geometric series with −1 < r < 1 Sn = a1 _

1 − r , r ≠ 1

number of permutations of n distinct objects taken r at a time P(n, r) = n! _ (n − r)!

number of combinations of n distinct objects taken r at a time C(n, r) = n! _ r!(n − r)!

number of permutations of n non-distinct objects n! _ r1!r2! … rk!

Binomial Theorem (x + y)n = ∑ k – 0

n

 n _ k  x n − ky k

(r + 1)th term of a binomial expansion  n _ r  xn − ryr

probability of an event with equally likely outcomes P(E)= n(E) ____ n(S)

probability of the union of two events P(E ∪ F) = P(E) + P(F) − P(E ∩ F)

probability of the union of mutually exclusive events P(E ∪ F) = P(E) + P(F)

probability of the complement of an event P(E ‘) = 1 − P(E)

1128 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

Key Concepts

13.1 Sequences and Their Notations • A sequence is a list of numbers, called terms, written in a specific order.

• Explicit formulas define each term of a sequence using the position of the term. See Example 1, Example 2, and Example 3.

• An explicit formula for the nth term of a sequence can be written by analyzing the pattern of several terms. See Example 4.

• Recursive formulas define each term of a sequence using previous terms.

• Recursive formulas must state the initial term, or terms, of a sequence.

• A set of terms can be written by using a recursive formula. See Example 5 and Example 6.

• A factorial is a mathematical operation that can be defined recursively.

• The factorial of n is the product of all integers from 1 to n See Example 7.

13.2 Arithmetic Sequences • An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.

• The constant between two consecutive terms is called the common difference.

• The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See Example 1.

• The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See Example 2 and Example 3.

• A recursive formula for an arithmetic sequence with common difference d is given by an = an − 1 + d, n ≥ 2. See Example 4.

• As with any recursive formula, the initial term of the sequence must be given.

• An explicit formula for an arithmetic sequence with common difference d is given by an = a1 + d(n − 1). See Example 5.

• An explicit formula can be used to find the number of terms in a sequence. See Example 6.

• In application problems, we sometimes alter the explicit formula slightly to an = a0 + dn. See Example 7.

13.3 Geometric Sequences • A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.

• The constant ratio between two consecutive terms is called the common ratio.

• The common ratio can be found by dividing any term in the sequence by the previous term. See Example 1.

• The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See Example 2 and Example 4.

• A recursive formula for a geometric sequence with common ratio r is given by an = ran − 1 for n ≥ 2 .

• As with any recursive formula, the initial term of the sequence must be given. See Example 3.

• An explicit formula for a geometric sequence with common ratio r is given by an = a1r n − 1. See Example 5.

• In application problems, we sometimes alter the explicit formula slightly to an = a0r n. See Example 6.

13.4 Series and Their Notations • The sum of the terms in a sequence is called a series.

• A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See Example 1.

• The sum of the terms in an arithmetic sequence is called an arithmetic series.

• The sum of the first n terms of an arithmetic series can be found using a formula. See Example 2 and Example 3.

• The sum of the terms in a geometric sequence is called a geometric series.

• The sum of the first n terms of a geometric series can be found using a formula. See Example 4 and Example 5.

• The sum of an infinite series exists if the series is geometric with −1 < r < 1.

CHAPTER 13 review 1129

• If the sum of an infinite series exists, it can be found using a formula. See Example 6, Example 7, and Example 8.

• An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See Example 9.

13.5 Counting Principles • If one event can occur in m ways and a second event with no common outcomes can occur in n ways, then the first

or second event can occur in m + n ways. See Example 1.

• If one event can occur in m ways and a second event can occur in n ways after the first event has occurred, then the two events can occur in m × n ways. See Example 2.

• A permutation is an ordering of n objects.

• If we have a set of n objects and we want to choose r objects from the set in order, we write P(n, r).

• Permutation problems can be solved using the Multiplication Principle or the formula for P(n, r). See Example 3 and Example 4.

• A selection of objects where the order does not matter is a combination.

• Given n distinct objects, the number of ways to select r objects from the set is C (n, r) and can be found using a formula. See Example 5.

• A set containing n distinct objects has 2n subsets. See Example 6.

• For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See Example 7.

13.6 Binomial Theorem

•  n _ r  is called a binomial coefficient and is equal to C (n, r). See Example 1. • The Binomial Theorem allows us to expand binomials without multiplying. See Example 2.

• We can find a given term of a binomial expansion without fully expanding the binomial. See Example 3.

13.7 Probability • Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is

certain.

• The probabilities in a probability model must sum to 1. See Example 1.

• When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See Example 2.

• To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See Example 3.

• To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See Example 4.

• The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See Example 5.

• In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See Example 6.

1130 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

ChAPTeR 13 RevIeW exeRCISeS

SeQUenCeS And TheIR nOTATIOn

1. Write the first four terms of the sequence defined by the recursive formula a1 = 2, an = an − 1 + n.

2. Evaluate 6! ________ (5 − 3)!3! .

3. Write the first four terms of the sequence defined by the explicit formula an = 10

n + 3. 4. Write the first four terms of the sequence defined by

the explicit formula an = n! ________ n(n + 1).

ARIThmeTIC SeQUenCeS

5. Is the sequence 4 _ 7 , 47 _ 21

, 82 _ 21

, 39 _ 7 , … arithmetic? If so,

find the common difference.

6. Is the sequence 2, 4, 8, 16, … arithmetic? If so, find the common difference.

7. An arithmetic sequence has the first term a1 = 18 and common difference d = −8. What are the first five terms?

8. An arithmetic sequence has terms a3 = 11.7 and a8 = −14.6. What is the first term?

9. Write a recursive formula for the arithmetic sequence −20, − 10, 0,10,…

10. Write a recursive formula for the arithmetic sequence 0, − 1 _

2 , −1, − 3 _

2 , … , and then find the 31st term.

11. Write an explicit formula for the arithmetic

sequence 7 _ 8

, 29 _ 24

, 37 _ 24

, 15 _ 8

, …

12. How many terms are in the finite arithmetic sequence 12, 20, 28, … , 172?

geOmeTRIC SeQUenCeS

13. Find the common ratio for the geometric sequence 2.5, 5, 10, 20, …

14. Is the sequence 4, 16, 28, 40, … geometric? If so find the common ratio. If not, explain why.

15. A geometric sequence has terms a7 = 16,384 and a9 = 262,144. What are the first five terms?

16. A geometric sequence has the first term a1 = −3 and

common ratio r = 1 _ 2

. What is the 8th term?

17. What are the first five terms of the geometric sequence a1 = 3, an = 4 ⋅ an − 1?

18. Write a recursive formula for the geometric

sequence 1, 1 _ 3

, 1 _ 9

, 1 _ 27

, …

19. Write an explicit formula for the geometric sequence

− 1 _ 5 , − 1 _

15 , − 1 _ 45 , −

1 _ 135

, …

20. How many terms are in the finite geometric

sequence −5, − 5 _ 3

, − 5 _ 9

, …, − 5 ______ 59,049 ?

SeRIeS And TheIR nOTATIOn

21. Use summation notation to write the sum of terms 1 _ 2

m + 5 from m = 0 to m = 5. 22. Use summation notation to write the sum that

results from adding the number 13 twenty times.

23. Use the formula for the sum of the first n terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series 2.5, 4, 5.5, … .

24. A ladder has 15 tapered rungs, the lengths of which increase by a common difference. The first rung is 5 inches long, and the last rung is 20 inches long. What is the sum of the lengths of the rungs?

CHAPTER 13 review 1131

25. Use the formula for the sum of the first n terms of a geometric series to find S9 for the series 12, 6, 3,

3 __ 2 , … 26. The fees for the first three years of a hunting club

membership are given in Table 1. If fees continue to rise at the same rate, how much will the total cost be for the first ten years of membership?

Year Membership Fees

1 \$1500

2 \$1950

3 \$2535 Table 1

27. Find the sum of the infinite geometric series

∑ k −1

45 ⋅   − 1 __ 3  k = 1

.

28. A ball has a bounce-back ratio 3 _ 5 of the height of

the previous bounce. Write a series representing the total distance traveled by the ball, assuming it was initially dropped from a height of 5 feet. What is the total distance? (Hint: the total distance the ball travels on each bounce is the sum of the heights of the rise and the fall.)

29. Alejandro deposits \$80 of his monthly earnings into an annuity that earns 6.25% annual interest, compounded monthly. How much money will he have saved after 5 years?

30. The twins Sarah and Scott both opened retirement accounts on their 21st birthday. Sarah deposits \$4,800.00 each year, earning 5.5% annual interest, compounded monthly. Scott deposits \$3,600.00 each year, earning 8.5% annual interest, compounded monthly. Which twin will earn the most interest by the time they are 55 years old? How much more?

COUnTIng PRInCIPleS

31. How many ways are there to choose a number from the set { −10, −6, 4, 10, 12, 18, 24, 32} that is divisible by either 4 or 6?

32. In a group of 20 musicians, 12 play piano, 7 play trumpet, and 2 play both piano and trumpet. How many musicians play either piano or trumpet?

33. How many ways are there to construct a 4-digit code if numbers can be repeated?

34. A palette of water color paints has 3 shades of green, 3 shades of blue, 2 shades of red, 2 shades of yellow, and 1 shade of black. How many ways are there to choose one shade of each color?

35. Calculate P(18, 4). 36. In a group of 5 freshman, 10 sophomores, 3 juniors, and 2 seniors, how many ways can a president, vice president, and treasurer be elected?

37. Calculate C(15, 6). 38. A coffee shop has 7 Guatemalan roasts, 4 Cuban roasts, and 10 Costa Rican roasts. How many ways can the shop choose 2 Guatemalan, 2 Cuban, and 3 Costa Rican roasts for a coffee tasting event?

39. How many subsets does the set {1, 3, 5, … , 99} have?

40. A day spa charges a basic day rate that includes use of a sauna, pool, and showers. For an extra charge, guests can choose from the following additional services: massage, body scrub, manicure, pedicure, facial, and straight-razor shave. How many ways are there to order additional services at the day spa?

41. How many distinct ways can the word DEADWOOD be arranged?

42. How many distinct rearrangements of the letters of the word DEADWOOD are there if the arrangement must begin and end with the letter D?

1132 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

bInOmIAl TheORem

43. Evaluate the binomial coefficient  23 _ 8  . 44. Use the Binomial Theorem to expand  3x + 1 _ 2

y  6 .

45. Use the Binomial Theorem to write the first three terms of (2a + b)17.

46. Find the fourth term of (3a 2 − 2b)11 without fully expanding the binomial.

PRObAbIlITy

For the following exercises, assume two die are rolled.

47. Construct a table showing the sample space. 48. What is the probability that a roll includes a 2?

49. What is the probability of rolling a pair? 50. What is the probability that a roll includes a 2 or results in a pair?

51. What is the probability that a roll doesn’t include a 2 or result in a pair?

52. What is the probability of rolling a 5 or a 6?

53. What is the probability that a roll includes neither a 5 nor a 6?

For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

54. What is the percent chance that all the children attending the party prefer soda?

55. What is the percent chance that at least one of the children attending the party prefers milk?

56. What is the percent chance that exactly 3 of the children attending the party prefer soda?

57. What is the percent chance that exactly 3 of the children attending the party prefer milk?

1133CHAPTER 13 Practice test

ChAPTeR 13 PRACTICe TeST

1. Write the first four terms of the sequence defined by

the recursive formula a = −14, an = 2 + an – 1 ________ 2 .

2. Write the first four terms of the sequence defined by

the explicit formula an = n2 − n − 1 ________ n! .

3. Is the sequence 0.3, 1.2, 2.1, 3, … arithmetic? If so find the common difference.

4. An arithmetic sequence has the first term a1 = −4

and common difference d = − 4 _ 3

. What is the 6th term?

5. Write a recursive formula for the arithmetic

sequence −2, − 7 _ 2

, − 5, − 13 _ 2

, … and then find the

22nd term.

6. Write an explicit formula for the arithmetic sequence 15.6, 15, 14.4, 13.8, … and then find the 32nd term.

7. Is the sequence − 2, − 1, − 1 _ 2

, − 1 _ 4 , … geometric? If

so find the common ratio. If not, explain why.

8. What is the 11th term of the geometric sequence − 1.5, − 3, − 6, − 12, … ?

9. Write a recursive formula for the geometric

sequence 1, − 1 _ 2

, 1 _ 4 , − 1 _ 8

, …

10. Write an explicit formula for the geometric sequence

4, − 4 _ 3

, 4 _ 9

, − 4 _ 27

, …

11. Use summation notation to write the sum of terms

3k 2 − 5 __ 6

k from k = −3 to k = 15.

12. A community baseball stadium has 10 seats in the first row, 13 seats in the second row, 16 seats in the third row, and so on. There are 56 rows in all. What is the seating capacity of the stadium?

13. Use the formula for the sum of the first n terms of a

geometric series to find ∑ k = 1

7

−0.2 ⋅ (−5)k − 1 .

14. Find the sum of the infinite geometric series.

∑ k = 1

1 _ 3

⋅  − 1 _ 5  k − 1

15. Rachael deposits \$3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned?

16. In a competition of 50 professional ballroom dancers, 22 compete in the fox-trot competition, 18 compete in the tango competition, and 6 compete in both the fox-trot and tango competitions. How many dancers compete in the foxtrot or tango competitions?

17. A buyer of a new sedan can custom order the car by choosing from 5 different exterior colors, 3 different interior colors, 2 sound systems, 3 motor designs, and either manual or automatic transmission. How many choices does the buyer have?

18. To allocate annual bonuses, a manager must choose his top four employees and rank them first to fourth. In how many ways can he create the “Top-Four” list out of the 32 employees?

19. A rock group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if have 15 songs to pick from?

20. A self-serve frozen yogurt shop has 8 candy toppings and 4 fruit toppings to choose from. How many ways are there to top a frozen yogurt?

21. How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E?

22. Use the Binomial Theorem to expand  3 _ 2 x − 1 _ 2

y  5 .

23. Find the seventh term of  x2 − 1 __ 2  13

without fully

expanding the binomial.

1134 CHAPTER 13 sequeNces, Probability aNd couNtiNg theory

For the following exercises, use the spinner in Figure 1.

Figure 1

1

2

3

4

5

6

7

24. Construct a probability model showing each possible outcome and its associated probability. (Use the first letter for colors.)

25. What is the probability of landing on an odd number?

26. What is the probability of landing on blue? 27. What is the probability of landing on blue or an odd number?

28. What is the probability of landing on anything other than blue or an odd number?

29. A bowl of candy holds 16 peppermint, 14 butterscotch, and 10 strawberry flavored candies. Suppose a person grabs a handful of 7 candies. What is the percent chance that exactly 3 are butterscotch? (Show calculations and round to the nearest tenth of a percent.)

C-1

Section 1.1 1. Irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational. 3. The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered. 5. −6 7. −2 9. −9 11. 9 13. −2 15. 4 17. 0 19. 9 21. 25 23. −6 25. 17 27. 4 29. −4 31. −6 33. ±1 35. 2 37. 2 39. −14y − 11 41. −4b + 1 43. 43z − 3 45. 9y + 45 47. −6b + 6 49. 16x _ 3

51. 9x 53. 1 _ 2 (40 − 10) + 5 55. Irrational number

57. g + 400 − 2(600) = 1200 59. Inverse property of addition 61. 68.4 63. True 65. Irrational 67. Rational

Section 1.2 1. No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 23 is the same as 2 × 2 × 2, which is 8. 32 is the same as 3 × 3, which is 9. 3. It is a method of writing very small and very large numbers.

5. 81 7. 243 9. 1 _ 16

11. 1 _ 11 13. 1 15. 4 9

17. 1240 19. 1 _ 79

21. 3.14 × 10−5 23. 16,000,000,000

25. a4 27. b 6 c 8 29. ab 2d 3 31. m4 33. q 5

_ p 6

35. y21

_ x14

37. 25 39. 72a 2 41. c 3 _

b 9 43.

y _

81z 6

45. 0.00135 m 47 1.0995 × 1012 49. 0.00000000003397 in.

51. 12,230,590,464 m 66 53. a 14 _

1296 55. n _

a 9c 57. 1 _

a 6b 6c 6

59. 0.000000000000000000000000000000000662606957

Section 1.3 1. When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1. 3. The principal square root is the nonnegative root of the number. 5. 16 7. 10 9. 14 11. 7 √

— 2

13. 9 √ — 5 _____ 5 15. 25 17. √

— 2 19. 2 √

— 6 21. 5 √

— 6

23. 6 √ —

35 25. 2 ___ 15

27. 6 √ —

10 ______ 19

29. − 1 + √ —

17 __________ 2

31. 7 3 √ — 2 33. 15 √

— 5 35. 20×2 37. 7 √

— p

39. 17m 2 √ —

m 41. 2b √ — a 43. 15x ___

7 45. 5y 4 √

— 2

47. 4 √ —

7d ______ 7d 49. 2 √

— 2 + 2 √

— 6x ____________ 1 − 3x 51. −w √

— 2w

53. 3 √ — x − √

— 3x ___________

2 55. 5n5 √

— 5 57. 9 √

— m _____

19m 59. 2 ___

3d

61. 3 4 √ —

2×2 ______ 2 63. 6z 3 √

— 2 65. 500 feet 67 −5 √

— 2  − 6 _________ 7

69. √ —

mnc ______ a9cmn 71. 2 √

— 2 x + √

— 2 ___________ 4 73.

√ — 3 ____ 3

Section 1.4

1. The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term. 3. Use the distributive property, multiply, combine like terms, and simplify. 5. 2 7. 8 9. 2 11. 4x 2 + 3x + 19 13. 3w 2 + 30w + 21 15. 11b4 − 9b3 + 12b2 − 7b + 8 17. 24x 2 − 4x − 8 19. 24b4 − 48b2 + 24 21. 99v 2 − 202v + 99 23. 8n3 − 4n2 + 72n − 36 25. 9y 2 − 42y + 49 27. 16p2 + 72p + 81 29. 9y 2 − 36y + 36 31. 16c 2 −1 33. 225n2 − 36 35. −16m2 + 16 37. 121q 2 −100 39. 16t 4 + 4t 3 − 32t 2 − t + 7 41. y 3 − 6y 2 − y + 18 43. 3p3 − p 2 − 12p + 10 45. a2 − b 2 47. 16t 2 − 40tu + 25u 2 49. 4t 2 + x 2 + 4t − 5tx − x 51. 24r 2 + 22rd − 7d 2 53. 32x 2 − 4x − 3m2 55. 32t 3 − 100t 2 + 40t + 38 57. a 4 + 4a3c − 16ac 3 − 16c4

Section 1.5 1. The terms of a polynomial do not have to have a common factor for the entire polynomial to be factorable. For example, 4×2 and −9y 2 don’t have a common factor, but the whole polynomial is still factorable: 4x 2 − 9y 2 = (2x + 3y)(2x − 3y). 3. Divide the x term into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression. 5. 7m 7. 10m3 9. y 11. (2a − 3)(a + 6) 13. (3n − 11)(2n + 1) 15. (p + 1)(2p − 7) 17. (5h + 3)(2h − 3) 19. (9d − 1)(d − 8) 21. (12t + 13)(t − 1) 23. (4x + 10)(4x − 10) 25. (11p + 13)(11p − 13) 27. (19d + 9)(19d − 9) 29. (12b + 5c)(12b − 5c) 31. (7n + 12)2 33. (15y + 4)2 35. (5p − 12)2 37. (x + 6)(x 2 − 6x + 36) 39. (5a + 7)(25a2 − 35a + 49) 41. (4x − 5)(16x 2 + 20x + 25) 43. (5r + 12s)(25r 2 − 60rs + 144s 2) 45. (2c + 3) −

1 _ 4 (−7c − 15)

47. (x + 2) − 2 _ 5 (19x + 10) 49. (2z − 9) −

3 ___ 2 (27z − 99) 51. (14x − 3)(7x + 9) 53. (3x + 5)(3x − 5) 55. (2x + 5)2 (2x − 5)2 57. (4z 2 + 49a2)(2z + 7a)(2z − 7a)

59. 1 ___ (4x + 9)(4x − 9)(2x + 3)

Section 1.6 1. You can factor the numerator and denominator to see if any of the terms can cancel one another out. 3. True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

5. y + 5 _____ y + 6

7. 3b + 3 9. x + 4 ______ 2x + 2

11. a + 3 _____ a − 3

13. 3n − 8 ______ 7n − 3

15. c − 6 _____ c + 6

17. 1 19. d 2 − 25 ________

25d 2 − 1

21. t + 5 _____ t + 3

23. 6x − 5 ______ 6x + 5

25. p + 6 ______ 4p + 3

27. 2d + 9 ______ d + 11

29. 12b + 5 _______ 3b−1

31. 4y − 1 ______ y + 4

33. 10x + 4y ________ xy

35. 9a − 7 __________ a2 − 2a − 3

37. 2y 2 − y + 9 __________

y 2 − y − 2 39. 5z

2 + z + 5 __________ z 2 − z − 2

41. x + 2xy + y _____________ x + xy + y + 1

43. 2b + 7a _______ ab2

45. 18 + ab _______ 4b

47. a − b 49. 3c 2 + 3c − 2 ___________

2c 2 + 5c + 2

51. 15x + 7 _______ x − 1

53. x + 9 _____ x − 9

55. 1 _____ y + 2

57. 4

Chapter 1 Review exercises 1. −5 3. 53 5. y = 24 7. 32m 9. Whole

11. Irrational 13. 16 15. a6 17. x 3

____ 32y 3

19. a

21. 1.634 × 107 23. 14 25. 5 √ — 3 27. 4 √

— 2 _____

5

29. 7 √ — 2 _ 50 31. 10 √

— 3 33. −3 35. 3x 3 + 4x 2 + 6

37. 5x 2 − x + 3 39. k2 − 3k − 18 41. x 3 + x 2 + x + 1 43. 3a2 + 5ab − 2b2 45. 9p 47. 4a2

49. (4a − 3)(2a + 9) 51. (x + 5)2 53. (2h − 3k)2

55. (p + 6)(p2 − 6p + 36) 57. (4q − 3p)(16q2 + 12pq + 9p2)

59. (p + 3) 1 _ 3 (−5p − 24) 61. x + 3 _ x − 4 63.

1 _ 2 65. m + 2 _ m − 3

67. 6x + 10y ________ xy

69. 1 _ 6

Chapter 1 practice test 1. Rational 3. x = 12 5. 3,141,500 7. 16

9. 9 11. 2x 13. 21 15. 3 √ — x _ 4 17. 21 √

— 6

19. 13q 3 − 4q 2 − 5q 21. n3 − 6n 2 + 12n − 8 23. (4x + 9)(4x − 9) 25. (3c − 11)(9c 2 + 33c + 121)

27. 4z − 3 ______ 2z − 1

29. 3a + 2b _______ 3b

ChapteR 2

Section 2.1 1. Answers may vary. Yes. It is possible for a point to be on the x-axis or on the y-axis and therefore is considered to NOT be in one of the quadrants. 3. The y-intercept is the point where the graph crosses the y-axis. 5. The x-intercept is (2, 0) and the y-intercept is (0, 6). 7. The x-intercept is (2, 0) and the y-intercept is (0, −3). 9. The x-intercept is (3, 0) and the

y-intercept is  0, 9 _ 8  . 11. y = 4 − 2x 13. y = 5 − 2x _________ 3

15. y = 2x − 4 _ 5 17. d = √ —

74 19. d = √ —

36 = 6

21. d ≈ 62.97 23.  3, − 3 _ 2  25. (2, −1) 27. (0, 0) 29. y = 0

31. Not collinear

x

y

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4 5

5

(−1, 2) (2, 1)

(0, 4)

33. (−3, 2), (1, 3), (4, 0)

35. x −3 0 3 6 y 1 2 3 4

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

(6, 4) (3, 3)

(0, 2)(−3, 1)

37. x −3 0 3 y 0 1.5 3

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

(3, 3)

(0, 1.5)(−3, 0)

39.

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

(8, 0)

(0, −4)

41.

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

(3, 0) (0, 2)

43. d = 8.246 45. d = 5 47. (−3, 4) 49. x = 0, y = −2 51. x = 0.75, y = 0 53. x = −1.667, y = 0 55. 15 − 11.2 = 3.8 mi shorter 57. 6.042 59. Midpoint of each diagonal is the same point (2, 2). Note this is a characteristic of rectangles, but not other quadrilaterals. 61. 37 mi 63. 54 ft

Section 2.2 1. It means they have the same slope. 3. The exponent of the x variable is 1. It is called a first-degree equation. 5. If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator). 7. x = 2

9. x = 2 _ 7 11. x = 6 13. x = 3 15. x = −14

17. x ≠ −4; x = −3 19. x ≠ 1; when we solve this we get x = 1, which is excluded, therefore NO solution 21. x ≠ 0; x = − 5 _ 2

23. y = − 4 ___ 5 x + 14 ___ 5

25. y = − 3 ___ 4 x + 2 27. y = 1 __ 2

x + 5 __ 2

29. y = −3x − 5 31. y = 7 33. y = −4 35. 8x + 5y = 7 37.

x

y

4

–2–4–6–8–10 –2 –4 –6 –8

–10

2

6

6

42 8

8

10

10

Parallel 39. Perpendicular

x

y

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

35.

1

x

y

−1

−2

−3

1 2 3 (t from −π to 0)

−1−2−3

2

3

37.

2

x

y

−2

−4

−6

2 4 6

(t from−π to 0)

−2−4−6

4

6

39. There will be 100 back-and-forth motions. 41. Take the opposite of the x(t) equation. 43. The parabola opens up.

45.  x(t) = 5 cos t y(t) = 5 sin t 47.

0.5

x

y

−0.5

−1

−1.5

0.5 1 1.5−0.5−1−1.5

1

1.5

49.

0.5

x

y

−0.5

−1

−1.5

0.5 1 1.5−0.5−1−1.5

1

1.5

(t from 0 to 2π)

51.

0.5

x

y

−0.5

−1

−15.

0.5 1 1.5−0.5−1−1.5

1

1.5

53. a = 4, b = 3, c = 6, d = 1

55. a = 4, b = 2, c = 3, d = 3

1 2 3

x

(t from 0 to 2π)

y

−2 −3 −4 −5 −6

−1 −0.5 0.5 1−1−1.5−2

4 5 6

57.

0.5 1

1.5

x

y

−1 −1.5

−2 −2.5

−3

−0.5 −1 1 2 3 4 5 6 7−2−3−4−5

2 2.5

3

(t from 0 to 2π) 0.5 1

1.5

x

y

−1 −1.5

−2 −2.5

−3

−0.5 −1 1 2 3 4 5 6 7−2−3−4−5

2 2.5

3

(t from 0 to 2π)

57. (cont.)

5

x

y

−5

−10

0.5 1−0.5−1−1.5−2

10

15

20

(t from −4π to 6π)

0.5

x

y

−0.5

−1.5

−2

−1

5 10 15 20−5−10

1

(t from −4π to 6π)

59.

0.5

x

y

−0.5

−1.5

−2

−1

5 10 15 20−5−10

1

(t from −4π to 6π)

61. The y-intercept changes.

63. y(x) = −16  x _ 15  2 + 20  x _ 15 

65.  x(t) = 64cos(52°) y(t) = −16t2 + 64tsin(52°) 67. Approximately 3.2 seconds 69. 1.6 seconds

71.

2 4 6

x

y

−4 −6 −8

−10 −12

−2 5 10−5−10

8 10 12

(t from 0 to 2π)

73.

2

x

y

−2

−4

−6

2 4 6

(t from 0 to 2π)

−2−4−6

4

6

Section 10.8 1. Lowercase, bold letter, usually u, v, w 3. They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1. 5. The first number always represents the coefficient of the i, and the second represents the j. 7. ⟨7, − 5⟩ 9. Not equal 11. Equal 13. Equal 15. −7i − 3j 17. −6i − 2j 19. u + v = ⟨−5, 5⟩, u − v = ⟨−1, 3⟩, 2u − 3v = ⟨0, 5⟩

21. −10i − 4j 23. − 2 √ —

29 _____ 29 i + 5 √

— 29 _____ 29 j

25. − 2 √ —

229 ______ 229 i + 15 √

— 229 ______ 229 j 27. −

7 √ — 2 ____ 10 i +

√ — 2 ____ 10 j

29. |v| = 7.810, θ = 39.806° 31. |v| = 7.211, θ = 236.310° 33. −6 35. −12 37. yy

x

y

v

x

3v

x

v12

39.

2u u + v

u − v

41.

u + v 2u

u − v

43. 45.

47. ⟨4, 1⟩ 49. v = −7i + 3j 51. 3 √

— 2 i + 3 √

— 2 j

53. i − √ — 3 j

55. a. 58.7; b. 12.5 57. x = 7.13 pounds, y = 3.63 pounds 59. x = 2.87 pounds, y = 4.10 pounds 61. 4.635 miles, 17.764° N of E 63. 17 miles, 10.071 miles 65. Distance: 2.868, Direction: 86.474° North of West, or 3.526° West of North 67. 4.924°, 659 km/hr 69. 4.424° 71. (0.081, 8.602) 73. 21.801°, relative to the car’s forward direction 75. Parallel: 16.28, perpendicular: 47.28 pounds 77. 19.35 pounds, 51.65° from the horizontal 79. 5.1583 pounds, 75.8° from the horizontal

Chapter 10 Review exercises 1. Not possible 3. C = 120°, a = 23.1, c = 34.1 5. Distance of the plane from point A: 2.2 km, elevation of the plane: 1.6 km 7. B = 71.0°, C = 55.0°, a = 12.8 9. 40.6 km 11. 13. (0, 2)

15. (9.8489, 203.96°) 17. r = 8 19. x 2 + y 2 = 7x

21. y = − x

1 2 3

x

y

−2 −1

−1 1 2 3 4 5 6 7−2−3−4−5−6−7

−4 −5 −6 −7

−3

4 5 6 7

23. Symmetric with respect to

the line θ = π __ 2

25.

1 2 3 4 5 6

(θ from 0 to 2π) 27.

1 2 3 4 5 6

(θ from 0 to 2π)

29. 5 31. cis  − π __ 3  33. 2.3 + 1.9i 35. 60cis  π __ 2 

37. 3cis  4π ___ 3  39. 25cis  3π ___ 2  41. 5cis 

3π ___ 4  , 5cis  7π ___ 4 

43.

1 2 3

Real

Imaginary

−2 −1

−1 1 2 3 4 5 6−2−3−4−5−6

−4 −5 −6

−3

4 5 6

45. x 2 + 1 __ 2 y = 1

47.  x(t) = −2 + 6t y(t) = 3 + 4t

49. y = −2x 5

x

y

−20

20

−40

−60

−80

−100

0.5 1.0 1.5 2.0

(t from −1 to 1)

51. a.

 x(t) = (80 cos (40°))t y(t) = − 16t 2 + (80 sin (40°))t + 4 b. The ball is 14 feet high and 184 feet from where it was launched. c. 3.3 seconds 53. Not equal 55. 4i

57. − 3 √ —

10 ______ 10 i, − √

— 10 _____ 10 j

59. Magnitude: 3 √ — 2 , Direction: 225°

61. 16 63.

3v

u + vu − v

Chapter 10 practice test 1. α = 67.1°, γ = 44.9°, a = 20.9 3. 1,712 miles 5.  1, √

— 3 

7. y = −3

1 2 3

x

y

−2 −3 −4 −5 −6

−1 −1 1 2 3 4 5 6−2−3−4−5−6

4 5 6

9.

1 2 3 4 5 6

(θ from 0 to 2π)

11. √ —

106 13. − 5 __ 2 + 5 √

— 3 _____ 2 i 15. 4cis (21°)

17. 2 √ — 2 cis(18°), 2 √

— 2 cis(198°) 19. y = 2(x − 1)2

21.

1 2 3

x

y

−2 −1

−1 1 2 3 4 5 6−2−3−4−5−6

−4 −5 −6

−3

4 5 6

(θ from 0 to 2π)

23. −4i − 15j

25. 2 √ —

13 ______ 13 i + 3 √

— 13 ______ 13 j

Chapter 11 Section 11.1 1. No, you can either have zero, one, or infinitely many. Examine graphs. 3. This means there is no realistic break-even point. By the time the company produces one unit they are already making profit. 5. You can solve by substitution (isolating x or y ), graphically, or by addition. 7. Yes 9. Yes 11. (−1, 2)

13. (−3, 1) 15.  − 3 _ 5 , 0  17. No solutions exist 19.  72 _ 5 ,

132 _ 5  21. (6, −6) 23.  − 1 __ 2 ,

1 ___ 10  25. No solutions exist. 27.  − 1 __ 5 ,

2 __ 3  29.  x, x + 3 _ 2 

31. (−4, 4) 33.  1 _ 2 , 1 _ 8  35. 

1 _ 6

, 0  37. ( x, 2(7x − 6)) 39.  − 5 __ 6 ,

4 __ 3  41. Consistent with one solution 43. Consistent with one solution 45. Dependent with infinitely many solutions 47. (−3.08, 4.91) 49. (−1.52, 2.29) 51.  A + B _ 2 ,

A − B _ 2  53.  − 1 _ A − B ,

A _ A − B

 55.  EC − BF _ AE − BD , DC − AF _ BD − AE

 57. They never turn a profit. 59. (1,250, 100,000) 61. The numbers are 7.5 and 20.5. 63. 24,000 65. 790 sophomores, 805 freshman 67. 56 men, 74 women 69. 10 gallons of 10% solution, 15 gallons of 60% solution 71. Swan Peak: \$750,000, Riverside: \$350,000 73. \$12,500 in the first account, \$10,500 in the second account 75. High- tops: 45, Low-tops: 15 77. Infinitely many solutions. We need more information.

Section 11.2 1. No, there can be only one, zero, or infinitely many solutions. 3. Not necessarily. There could be zero, one, or infinitely many solutions. For example, (0, 0, 0) is not a solution to the system below, but that does not mean that it has no solution. 2x + 3y − 6z = 1 −4x − 6y + 12z = −2 x + 2y + 5z = 10 5. Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable. 7. No 9. Yes

11. (−1, 4, 2) 13.  − 85 ____ 107 , 312 ____ 107 ,

191 ____ 107  15.  1, 1 _ 2 , 0 

17. (4, −6, 1) 19.  x, 65 − 16x _______ 27 , 28 + x ______ 27  21.  −

45 ___ 13 , 17 ___ 13 , −2 

23. No solutions exist 25. (0, 0, 0) 27.  4 __ 7 , − 1 __ 7 , −

3 __ 7  29. (7, 20, 16) 31. (−6, 2, 1) 33. (5, 12, 15)

35. (−5, −5, −5) 37. (10, 10, 10) 39.  1 __ 2 , 1 __ 5

, 4 __ 5

 41.  1 __ 2 ,

2 __ 5

, 4 __ 5

 43. (2, 0, 0) 45. (1, 1, 1) 47.  128 ____ 557 ,

23 ____ 557

, 28 ____ 557

 49. (6, −1, 0) 51. 24, 36, 48 53. 70 grandparents, 140 parents, 190 children 55. Your share was \$19.95, Sarah’s share was \$40, and your other roommate’s share was \$22.05.

57. There are infinitely many solutions; we need more information. 59. 500 students, 225 children, and 450 adults 61. The BMW was \$49,636, the Jeep was \$42,636, and the Toyota was \$47,727. 63. \$400,000 in the account that pays 3% interest, \$500,000 in the account that pays 4% interest, and \$100,000 in the account that pays 2% interest. 65. The United States consumed 26.3%, Japan 7.1%, and China 6.4% of the world’s oil. 67. Saudi Arabia imported 16.8%, Canada imported 15.1%, and Mexico 15.0% 69. Birds were 19.3%, fish were 18.6%, and mammals were 17.1% of endangered species

Section 11.3 1. A nonlinear system could be representative of two circles that overlap and intersect in two locations, hence two solutions. A nonlinear system could be representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches also intersect the circle, hence three solutions. 3. No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region. 5. Choose any number between each solution and plug into C(x) and R(x). If C(x) < R(x), then there is profit.

7. (0, −3), (3, 0) 9.  − 3 √ — 2 _____

2 , 3 √

— 2 _____

2  ,  3 √

— 2 _____

2 , − 3 √

— 2 _____

2 

11. (−3, 0), (3, 0) 13.  1 __ 4 , − √

— 62 _____

8  ,  1 __ 4 ,

√ —

62 _____ 8

15.  − √ —

398 ______ 4

, 199 ____ 4

 ,  √ —

398 ______ 4

, 199 ____ 4

 17. (0, 2), (1, 3)

19.  − √ __________

1 __ 2 ( √ — 5 − 1) , 1 __ 2 (1 − √

— 5 )  ,  √

__________

1 __ 2 ( √ — 5 − 1) , 1 __ 2 (1 − √

— 5 ) 

21. (5, 0) 23. (0, 0) 25. (3, 0) 27. No solutions exist 29. No solutions exist

31.  − √ — 2 ____

2 , − √

— 2 ____

2  ,  − √

— 2 ____

2 , √

— 2 ____

2  ,  √

— 2 ____

2 , − √

— 2 ____

2  ,  √

— 2 ____

2 , √

— 2 ____

2 

33. (2, 0) 35. (− √— 7 , −3), (− √— 7 , 3), ( √— 7 , −3), ( √— 7 , 3)

37.

39.

2 4 6

x

y

−4 −2−1 1 2 3 4 5−2−3−4−5

−8 −10

−6

8 10

41.

1 2 3

x

y

−2 −1−1 1 2 3 4 5−2−3−4−5

−4 −5

−3

4 5

 − √ ___________

1 _ 2 ( √ —

73 −5) , 1 _ 2 (7 − √ —

73 )   √ ___________

1 __ 2 ( √ —

73 − 5) , 1 __ 2 (7 − √ —

73 ) 

43. 45.

2 4 6

x

y

−4 −2−2 2 4 6 8−4−6−8

−8 −10

−6

8

47.

1 2 3

x

y

−2 −1−1 1 2 3 4 5−2−3−4−5

−4 −5

−3

4 5 49.  −2 √

____

70 ____ 383

, −2 √ ___

35 ___ 29

 ,  −2 √

____

70 ____ 383 , 2 √ ___

35 ___ 29  ,  2 √

____

70 ____ 383 , −2 √ ___

35 ___ 29  ,  2 √

____

70 ____ 383

, 2 √ ___

35 ___ 29

 51. No solution exists 53. x = 0, y > 0 and 0 < x < 1, √

— x < y < 1 _ x 55. 12,288

57. 2–20 computers

Section 11.4 1. No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, 1 _

x 2 + 1 cannot be

decomposed because the denominator cannot be factored. 3. Graph both sides and ensure they are equal. 5. If we choose x = −1, then the B-term disappears, letting us immediately know that A = 3. We could alternatively plug in x = − 5 _ 3 giving us a B-value of −2.

7. 8 _____ x + 3

− 5 _____ x − 8

9. 1 _____ x + 5

+ 9 _____ x + 2

11. 3 ______ 5x − 2

+ 4 ______ 4x − 1

13. 5 ________ 2(x + 3)

+ 5 _______ 2(x − 3)

15 3 _____ x + 2

+ 3 _____ x − 2

17. 9 ________ 5(x + 2)

+ 11 ________ 5(x − 3)

19. 8 _____ x − 3

− 5 _____ x − 2

21. 1 _____ x − 2

+ 2 _______ (x − 2) 2

23. − 6 ______ 4x + 5

+ 3 ________ (4x + 5) 2

25. − 1 _____ x − 7

− 2 _______ (x − 7) 2

27. 4 __ x

− 3 ________ 2(x + 1)

+ 7 ________ 2(x + 1) 2

29. 4 __ x

+ 2 ___ x 2

− 3 ______ 3x + 2

+ 7 _________ 2(3x + 2) 2

31. x + 1 _________ x 2 + x + 3

+ 3 _____ x + 2

33. 4 − 3x __________ x 2 + 3x + 8

+ 1 _____ x − 1

35. 2x − 1 __________ x 2 + 6x + 1

+ 2 _____ x + 3

37. 1 _________ x 2 + x + 1

+ 4 _____ x − 1

39. 2 _________ x 2 −3x + 9

+ 3 _____ x + 3

41. − 1 ___________ 4x 2 + 6x + 9

+ 1 ______ 2x − 3

43. 1 __ x

+ 1 _____ x + 6

− 4x ___________ x 2 − 6x + 36 45. x + 6 ______ x 2 + 1

+ 4x + 3 _______ (x 2 + 1) 2

47. x + 1 _____ x + 2

+ 2x + 3 _______ (x + 2) 2

49. 1 ___________ x 2 + 3x + 25

− 3x _____________ (x 2 + 3x + 25) 2

51. 1 ___ 8x

− x ________ 8(x 2 + 4)

+ 10 − x _________ 2(x 2 + 4) 2

53. − 16 ___ x

− 9 ___ x 2

+ 16 _____ x − 1

− 7 _______ (x − 1) 2

55. 1 _____ x + 1

− 2 _______ (x + 1) 2

+ 5 _______ (x + 1)3

57. 5 _____ x − 2

− 3 _________ 10(x + 2)

+ 7 _____ x + 8

− 7 _________ 10(x − 8)

59. − 5 ___ 4x

− 5 ________ 2(x + 2)

+ 11 ________ 2(x + 4)

+ 5 _______ 4(x − 4)

Section 11.5 1. No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a 2×2 matrix and the

second is a 2×3 matrix.  1 2 3 4  +  6 5 4

3 2 1

 has no sum. 3. Yes, if the dimensions of A are m × n and the dimensions of B are n × m, both products will be defined 5. Not necessarily. To find AB, we multiply the first row of A by the first column of B to get the first entry of AB. To find BA, we multiply the first row of B by the first column of A to get the first entry of BA. Thus, if those are unequal, then the matrix multiplication does not commute.

7.  11 19

15 94 17 67

 9.  −4 2 8 1  11. Undefined; dimensions do not match

13.  9 27

63 36 0 192

 15.  −64 −12 −28 −72 −360 −20 −12 −116 

17. 

1,800 1,200 1,300

800 1,400 600

700 400 2,100  19.  20 102 28 28 

21.  60 41 2 −16 120 −216  23.  −68 24 136

−54 −12 64 −57 30 128

 25. Undefined; dimensions do not match.

27.  −8 41 −3

40 −15 −14 4 27 42

 29.  −840 650 −530

330 360 250 −10 900 110

 31.  −350 1,050 350 350  33. Undefined; inner dimensions do not match.

35.  1,400 700 −1,400 700  37.  332,500 927,500

−227,500 87,500 

39.  490,000 0 0 490,000  41.  −2 3 4 −7 9 −7

43.  −4 29 21 −27 −3 1  45.  −3 −2 −2

−28 59 46 −4 16 7

 47. 

1 −18 −9

−198 505 369

−72 126 91  49.  0 1.6 9 −1 

51.  2 24 −4.5

12 32 −9 −8 64 61

 53.  0.5 3 0.5

2 1 2 10 7 10

 55. 

1 0 0

0 1 0

0 0 1  57. 

1 0 0

0 1 0

0 0 1 

2 4 6

x

y

−4 −2−2 2 4 6 8 10−4−6−8−10

−8 −10

−6

8 10

59. Bn =   1 0 0

0 1 0 0 0 1

 , n even, 

1 0 0

0 0 1

0 1 0  , n odd.

Section 11.6 1. Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar. 3. No, there are numerous correct methods of using row operations on a matrix. Two possible ways are the following: (1) Interchange rows 1 and 2. Then R 2 = R 2 − 9R 1. (2) R 2 = R 1 −9R 2. Then divide row 1 by 9. 5. No. A matrix with 0 entries for an entire row would have either zero or infinitely many solutions.

7.  0 16 9 −1 | 4 2

 9.  1 5 8

12 3 0 3 4 9

| 19

4 −7

 11. −2x + 5y = 5 6x − 18y = 26

13. 3x + 2y = 13 −x − 9y + 4z = 53 8x + 5y + 7z = 80

15. 4x + 5y − 2z = 12 y + 58z = 2 8x + 7y − 3z = −5

17. No solutions 19. (−1, −2) 21. (6, 7)

23. (3, 2) 25.  1 __ 5 , 1 __ 2

 27.  x, 4 ___ 15 (5x + 1)  29. (3, 4) 31.  196 ____ 39 , −

5 ___ 13  33. (31, −42, 87) 35.  21 ___ 40

, 1 ___ 20

, 9 __ 8

 37.  18 ___ 13 ,

15 ___ 13 , − 15 ___ 13  39.  x, y,

1 _ 2 − x − 3 _ 2 y 

41.  x, − x __ 2 , −1  43. (125, −25, 0) 45. (8, 1, −2) 47. (1, 2, 3) 49.  −4z + 17 __ 7 , 3z −

10 __ 7 , z  51. No solutions exist. 53. 860 red velvet, 1,340 chocolate 55. 4% for account 1, 6% for account 2 57. \$126 59. Banana was 3%, pumpkin was 7%, and rocky road was 2% 61. 100 almonds, 200 cashews, 600 pistachios

Section 11.7 1. If A−1 is the inverse of A, then AA−1 = I, the identity matrix. Since A is also the inverse of A−1, A−1 A = I. You can also check by proving this for a 2 × 2 matrix. 3. No, because ad and bc are both 0, so ad − bc = 0, which requires us to divide by 0 in the

formula. 5. Yes. Consider the matrix  0 1 1 0  . The inverse is found with the following calculation: A−1 = 1 __________

0(0) − 1(1)  0 −1 −1 0  = 

0 1

1 0  .

7. AB = BA =  1 0 0 1  = I 9. AB = BA =  1 0

0 1

 = I

11. AB = BA =  1 0 0

0 1 0 0 0 1

 = I 13. 1 ___ 29  9 2 −1 3  15. 1 ___

69  −2 7 9 3  17. There is no inverse 19.

4 __ 7

 0.5 1.5 1 −0.5 

21. 1 ___

17 

−5 5 −3

20 −3 12

1 −1 4  23. 1 ____ 209 

47 −57 69

10 19 −12

−24 38 −13 

25.  18 60 −168

−56 −140 448 40 80 −280

 27. (−5, 6) 29. (2, 0) 31.  1 __ 3 , −

5 __ 2

 33.  − 2 __ 3 , − 11 ___ 6  35.  7,

1 __ 2

, 1 __ 5

 37. (5, 0, −1) 39.  − 35 __ 34 , −

97 __ 34 , − 77 __ 17 

41.  13 _ 138 , − 568 ___ 345 , −

229 _ 690

 43.  − 37 ___ 30 , 8 ___

15 

45.  10 ____ 123 , −1, 2 __ 5

 47. 1 __ 2  2 1 −1 −1

0 1 1 −1 0 −1 1 1

0 1 −1 1 

49. 1 ___ 39

 3 2 1 −7

18 −53 32 10 24 −36 21 9

−9 46 −16 −5 

51.  1 0 0 0 0 0

0 1 0 0 0 0 0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0 −1 −1 −1 −1 −1 1

 53. Infinite solutions 55. 50% oranges, 25% bananas, 20% apples 57. 10 straw hats, 50 beanies, 40 cowboy hats 59. Tom ate 6, Joe ate 3, and Albert ate 3 61. 124 oranges, 10 lemons, 8 pomegranates

Section 11.8 1. A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product—even if it does end up being 0. 3. The inverse does not exist. 5. −2 7. 7 9. −4 11. 0 13. −7, 990.7 15. 3 17. −1 19. 224 21. 15 23. −17.03 25. (1, 1) 27.  1 __ 2 ,

1 __ 3

 29. (2, 5) 31.  −1, − 1 __ 3  33. (15, 12) 35. (1, 3, 2) 37. (−1, 0, 3) 39.  1 __ 2 , 1, 2  41. (2, 1, 4) 43. Infinite solutions 45. 24 47. 1 49. Yes; 18, 38 51. Yes; 33, 36, 37 53. \$7,000 in first account, \$3,000 in second account 55. 120 children, 1,080 adult 57. 4 gal yellow, 6 gal blue 59. 13 green tomatoes, 17 red tomatoes 61. Strawberries 18%, oranges 9%, kiwi 10% 63. 100 for the first movie, 230 for the second movie, 312 for the third movie 65. 20–29: 2,100, 30–39: 2,600, 40–49: 825 67. 300 almonds, 400 cranberries, 300 cashews

Chapter 11 Review exercises 1. No 3. (−2, 3) 5. (4, −1) 7. No solutions exist 9. (300, 60) 11. (10, −10, 10) 13. No solutions exist

15. (−1, − 2, 3) 17.  x, 8x ___ 5 , 14x ____ 5  19. 11, 17, 33

21. (2, − 3), (3, 2) 23. No solution 25. No solution

27.

1 2 3

x

y

−2 −1−1 1 2 3 4 5−2−3−4−5

−4 −5

−3

4 5

29.

1 2 3

x

y

−2 −1−1 1 2 3 4 5 6 7−2−3

−4 −5

−3

4 5

31. − 10 _ x + 2 + 8 _ x + 1 33.

7 _ x + 5 − 15 _

(x + 5)2

35. 3 _ x − 5 + −4x + 1 __

x2 + 5x + 25 37. x − 4 _

x 2 − 2 + 5x + 3 _

(x 2 − 2)2

39.  −16 8 −4 −12  41. Undefined; dimensions do not match 43. Undefined; inner dimensions do not match

45.  113 28 10

44 81 −41 84 98 −42

 47.  −127 −74 176

−2 11 40 28 77 38

 49. Undefined; inner dimensions do not match 51. x − 3z = 7 y + 2z = − 5 with infinite solutions

53.  −2 2 1

2 −8 5 19 −10 22

| 7

0 3

 55.  1 0 3

−1 4 0 0 1 2

| 12

0 −7

 57. No solutions exist 59. No solutions exist

61. 1 __ 8

 2 7 6 1  63. No inverse exists 65. (−20, 40) 67. (−1, 0.2, 0.3) 69. 17% oranges, 34% bananas, 39% apples

71. 0 73. 6 75.  6, 1 __ 2  77. ( x, 5x + 3) 79.  0, 0, − 1 __ 2

Chapter 11 practice test 1. Yes 3. No solutions exist 5.  1 _ 2 ,

1 _ 4 , 1 _ 5 

7.  x, 16x ____ 5 , − 13x ____ 5 

9. (−2 √— 2 , − √— 17 ), (−2 √— 2 , √— 17 ), (2 √— 2 , − √— 17 ), (2 √— 2 , √— 17 ) 11.

1 2 3

x

y

−2 −1−1 1 2 3 4 5−2−3−4−5

−4 −5

−3

4 5

13. 5 ______ 3x + 1

− 2x + 3 ________ (3x + 1)2

15.  17 51 −8 11 

17.  12 −20 −15 30 

19. − 1 __ 8

21.  14 −2 13

−2 3 −6 1 −5 12

| 140

−1 11

 23. No solutions exist. 25. (100, 90) 27.  1 ____ 100 , 0  29. 32 or more cell phones per day

ChapteR 12

Section 12.1 1. An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

3. This special case would be a circle. 5. It is symmetric about

the x-axis, y-axis, and the origin.

7. Yes; x 2

___ 32

+ y 2

___ 22

= 1 9. Yes; x 2 ______

 1 __ 2  2 +

y 2 ______

 1 __ 3  2 = 1

11. x 2

___ 22

+ y 2

___ 72

= 1; endpoints of major axis: (0, 7) and (0, −7); endpoints

of minor axis: (2, 0) and (−2, 0); foci: (0, 3 √— 5 ), (0, −3 √— 5 ) 13. x

2 _

(1)2 +

y 2 _

 1 __ 3  2 = 1; endpoints of major axis: (1, 0) and (−1, 0);

endpoints of minor axis:  0, 1 _ 3  ,  0, − 1 _ 3  ; foci:  2 √

— 2 _ 3 , 0  ,

 − 2 √ — 2 _ 3 , 0  15.

(x − 2)2 _

72 +

(y − 4)2 _

52 = 1; endpoints of major

axis: (9, 4), (−5, 4); endpoints of minor axis: (2, 9), (2, − 1); foci:

(2 + 2 √— 6 , 4), (2 − 2 √— 6 , 4) 17. (x + 5) 2

_ 22 + (y − 7)2

_ 32 = 1;

endpoints of major axis: (−5, 10), (−5, 4); endpoints of minor axis:

(−3, 7), (−7, 7); foci: (−5, 7 + √— 5 ), (−5, 7 − √— 5 )

19. (x − 1)2

_ 32 + (y − 4)2

_ 22 = 1; endpoints of major axis: (4, 4), (−2, 4);

endpoints of minor axis: (1, 6), (1, 2); foci: (1 + √— 5 , 4), (1 − √— 5 , 4) 21. (x − 3)

2

_  3 √

— 2  2

+ (y − 5)2

_  √

— 2  2

= 1; endpoints of

major axis: (3 + 3 √— 2 , 5 ), (3 − 3 √— 2 , 5); endpoints of minor axis: (3, 5 + √— 2 ), (3, 5 − √— 2 ); foci: (7, 5), (−1, 5)

23. (x + 5) 2 _______ 52 +

(y − 2)2 _______ 22 = 1; endpoints of major axis: (0, 2), (−10, 2);

endpoints of minor axis: (−5, 4), (−5, 0); foci: (−5 + √— 21 , 2), (−5 − √— 21 , 2) 25. (x + 3)2 _______ 52 +

(y + 4)2 _______ 22 = 1; endpoints of

major axis (2, −4), (−8, −4); endpoints of minor axis (−3, −2),

(−3, − 6); foci: (−3 + √— 21 , −4), (−3 − √— 21 , −4). 27. Foci: (−3, −1 + √— 11 ), (−3, −1 − √— 11 ) 29. Focus: (0, 0) 31. Foci: (−10, 30), (−10, −30)

33. Center: (0, 0); vertices: (4, 0), (−4, 0), (0, 3), (0, −3);

foci: ( √— 7 , 0), (− √— 7 , 0)

1 2 3

x

y

−2 −1−1 1 2 3 4 5−2−3−4−5

−4 −5

−3

4 5

35. Center (0, 0); vertices:  1 __ 9 , 0  ,  − 1 __ 9 , 0  ,  0,

1 __ 7

 ,  0, − 1 __ 7  ; foci  0, 4 √

— 2 _____

63  ,  0, − 4 √

— 2 _____

63 

.1

.2

.3

x

y

−.2 −.1−.1 .1 .2 .3−.2−.3

−.3

Chapter 12 practice test

1. x 2

___ 32

+ y 2

___ 22

= 1; center: (0, 0); vertices: (3, 0), (−3, 0), (0, 2),

(0, −2); foci: ( √— 5 , 0), (− √— 5 , 0) 3. Center: (3, 2); vertices: (11, 2), (−5, 2), (3, 8), (3, −4); foci: (3 + 2 √— 7 , 2), (3 − 2 √— 7 , 2)

5 10 15

x

y

−10 −5 −5 5 10 15−10−15

−15

5. (x − 1) 2

_______ 36

+ (y −2) 2

______ 27

= 1

7. x 2 _

72 −

y 2 _

92 = 1; center: (0, 0);

vertices (7, 0), (−7, 0);

foci: ( √— 130 , 0), (− √— 130 , 0); asymptotes: y = ± 9 __

7 x

9. Center: (3, −3); vertices: (8, −3), (−2, −3); foci: (3 + √— 26 , −3), (3 − √— 26 , −3); asymptotes: y = ± 1 _ 5 (x − 3) − 3

5

x

y

−5

5 10 15 20−5−10−15−20

−10

Vertex (−2, −3)

Vertex (8, −3)

Center (3, −3) Focus

Focus

11. (y − 3) 2

_______ 1

− (x − 1) 2

_______ 8

= 1 13. (x − 2)2 = 1 _ 3 (y + 1);

vertex: (2, −1); focus:  2, − 11 ___ 12  ; directrix: y = − 13 ___ 12

15.

5

10

15

x

y

Focus (−5, 4)

x = −1−5

−10

5 10−5−10−15−20

Vertex (−3, 4)

17. Approximately 8.48 feet 19. Parabola; θ ≈ 63.4°

21. x′ 2 − 4x′ + 3y′ = 0

1 2 3

x

y

−2 −1−1 1 2 3 4 5 6−2−3−4−5−6

−4 −5 −6

−3

 2,

 3

4

23. Hyperbola with e = 3 _ 2 , and directrix 5 _

6 units to the

right of the pole.

25.

0.4

0.8

x

y

−0.4

0.4 0.8 1.2 1.6−0.4−0.8−1.2−1.6

−0.8

−1.2

−1.6

−2

Vertex 0,  4

1

y = 2 1

Focus (0, 0)

ChapteR 13

Section 13.1 1. A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers. 3. Yes, both sets go on indefinitely, so they are both infinite sequences. 5. A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out 13 ∙ 12 ∙ 11 ∙ 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1.

7. First four terms: −8, − 16 _ 3 , − 4, − 16 _ 5 9. First four terms:

2, 1 _ 2 , 8 _ 27 ,

1 _ 4 11. First four terms: 1.25, −5, 20, −80

13. First four terms: 1 _ 3 , 4 _ 5 ,

9 _ 7 , 16 _ 9 15. First four terms:

− 4 _ 5 , 4, −20, 100 17. 1 _ 3 ,

4 _ 5 , 9 _ 7 ,

16 _ 9 , 25 _ 11 , 31, 44, 59

19. −0.6, −3, −15, −20, −375, −80, −9375, −320

21. an = n 2 + 3 23. an =

2n _ 2n or 2n − 1 _ n 25. an =  − 1 _ 2 

n − 1

27. First five terms: 3, −9, 27, −81, 243 29. First five terms:

−1, 1, −9, 27 _ 11 , 891 _ 5 31.

1 _ 24 , 1, 1 _ 4 ,

3 _ 2 , 9 _ 4 ,

81 _ 4 , 2187 _ 8 ,

531,441 _ 16

33. 2, 10, 12, 14 _ 5 , 4 _ 5 , 2, 10, 12 35. a 1 = −8, an = an − 1 + n

37. a 1 = 35, an = an − 1 + 3 39. 720 41. 665,280

43. First four terms: 1, 1 _ 2 , 2 _ 3 ,

3 _ 2

45. First four terms: −1, 2, 6 _ 5 , 24 _ 11

47.

1 2 3

n

an

−2 −1−1 1 2 3 4 5 6−2−3−4−5−6

−4 −5 −6

−3

4 5 6

(1, 0)

49.

1 2 3

n

an

−2 −1−1 1 2 3 4 5 6−2−3−4−5−6

−4 −5 −6

−3

4 5 6

(1, 2)

(5, 0)

(4, 1)(2, 1)

(3, 0)

51.

6 12 18

n

an

−12 −6−1 1

(1, 2)

(2, 6) (3, 12)

(4, 20) (5, 30)

2 3 4 5 6 7−2−3−4−5

−24 −30 −36

−18

24 30 36

53. an = 2 n − 2

55. a 1 = 6, an = 2an − 1 − 5

57. First five terms:

29 ___ 37

, 152 ____ 111

, 716 ____ 333

, 3188 _____ 999

, 13724 ______ 2997

59. First five terms: 2, 3, 5, 17, 65537 61. a 10 = 7,257,600

63. First six terms: 0.042, 0.146, 0.875, 2.385, 4.708 65. First four terms: 5.975, 32.765, 185.743, 1057.25, 6023.521

67. If an = −421 is a term in the sequence, then solving the equation −421 = −6 − 8n for n will yield a non-negative integer. However, if −421 = −6 − 8n, then n = 51.875 so an = −421 is not a term in the sequence. 69. a 1 = 1, a 2 = 0, an = an − 1 − an − 2

71. (n + 2)! _______ (n − 1)!

= (n + 2) · (n + 1) · (n) · (n − 1) · … · 3 · 2 · 1 ___________________________________ (n − 1) · … · 3 · 2 · 1

= n(n + 1)(n + 2) = n 3 + 3n 2 + 2n

Section 13.2

1. A sequence where each successive term of the sequence increases (or decreases) by a constant value. 3. We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference. 5. Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers. 7. The common difference is 1 _ 2 9. The sequence is not arithmetic because

16 − 4 ≠ 64 − 16. 11. 0, 2 _ 3 , 4 _ 3 , 2,

8 _ 3 13. 0, −5, −10, −15, −20

15. a 4 = 19 17. a 6 = 41 19. a 1 = 2 21. a 1 = 5 23. a 1 = 6 25. a 21 = −13.5 27. −19, −20.4, −21.8, −23.2, −24.6 29. a 1 = 17; an = an − 1 + 9; n ≥ 2 31. a 1 = 12; an = an − 1 + 5; n ≥ 2

33. a 1 = 8.9; an = an − 1 + 1.4; n ≥ 2 35. a 1 = 1 __ 5 ; an = an − 1 +

1 __ 4 ; n ≥ 2

37. a 1 = 1 __ 6 ; an = an − 1 −

13 ___ 12 ; n ≥ 2 39. a 1 = 4; an = an − 1 + 7; a 14 = 95

41. First five terms: 20, 16, 12, 8, 4 43. an = 1 + 2n 45. an = −105 + 100n 47. an = 1.8n 49. an = 13.1 + 2.7n

51. an = 1 _ 3 n −

1 _ 3 53. There are 10 terms in the sequence.

55. There are 6 terms in the sequence. 57. The graph does not represent an arithmetic sequence.

59.

5 10

n

an

−5 −10 −15 −20 −25 −30 −35

−0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

(1, 9) (2, −1)

(3, −11)

(4, −21)

(5, −31)

65.

1 2 3

n

an

−1−1 1 2 3 4 5 6−2−3−4−5−6

4 5 6 7 8 9

10

(1, 5.5) (3, 6.5)

(4, 7) (4, 7.5) (2, 6)

61. 1, 4, 7, 10, 13, 16, 19 63.

1 2 3

n

an

−1−1 1 2 3 4 5 6−2−3−4−5−6

4 5 6 7 8 9

10 11 12 13 14

(1, 1)

(2, 4)

(3, 7)

(4, 10)

(5, 13)

67. Answers will vary. Examples: an = 20.6n and an = 2 + 20.4n 69. a 11 = −17a + 38b

71. The sequence begins to have negative values at the 13th

term, a 13 = − 1 _ 3 73. Answers will vary. Check to see that the

sequence is arithmetic. Example: Recursive formula: a 1 = 3, an = an − 1 − 3. First 4 terms: 3, 0, −3, −6; a 31 = −87

Section 13.3 1. A sequence in which the ratio between any two consecutive terms is constant. 3. Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric. 5. Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive. 7. The common ratio is −2 9. The sequence is geometric. The common ratio is 2. 11. The sequence is geometric. The

common ratio is − 1 _ 2 . 13. The sequence is geometric. The

common ratio is 5. 15. 5, 1, 1 _ 5 , 1 _ 25 ,

1 _ 125 17. 800, 400, 200, 100, 50

19. a 4 = − 16 _ 27 21. a 7 = −

2 _ 729 23. 7, 1.4, 0.28, 0.056, 0.0112

25. a = −32, an = 1 _ 2 an − 1 27. a 1 = 10, an = −0.3 an − 1

29. a 1 = 3 _ 5 , an =

1 _ 6

an − 1 31. a 1 = 1 _ 512 , an = −4an − 1

33. 12, −6, 3, − 3 _ 2 , 3 _ 4 35. an = 3

n − 1

37. an = 0.8 ∙ (−5) n − 1 39. an = −  4 __ 5 

n − 1

41. an = 3 ∙  − 1 __ 3  n − 1

43. a 12 = 1 ________

177, 147

45. There are 12 terms in the sequence. 47. The graph does not represent a geometric sequence. 49.

12 24 36

n

an

−1−0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

(1, 3) (2, 6) (3, 12) (4, 24)

(5, 48) 48 60

51. Answers will vary. Examples: a 1 = 800, an = 0.5an − 1 and a 1 = 12.5, an = 4an − 1 53. a 5 = 256b 55. The sequence exceeds 100 at the 14th term, a 14 ≈ 107.

57. a 4 = − 32 ___ 3

is the first non-integer value

59. Answers will vary. Example: explicit formula with a decimal common ratio: an = 400 ∙ 0.5

n − 1; first 4 terms: 400, 200, 100, 50; a 8 = 3.125

Section 13.4 1. An nth partial sum is the sum of the first n terms of a sequence. 3. A geometric series is the sum of the terms in a geometric sequence. 5. An annuity is a series of regular equal payments that earn a constant compounded interest.

7. ∑ n = 0

4

5n 9. ∑ k = 1

5

4 11. ∑ k = 1

20

8k + 2 13. S 5 = 25 _ 2

15. S 13 = 57.2 17. ∑ k = 1

7

8 ∙ 0.5k − 1

19. S 5 = 9  1 −  1 _ 3 

5  __

1 − 1 _ 3 = 121 _ 9 ≈ 13.44

21. S 11 = 64(1 − 0.211)

__ 1 − 0.2 = 781,249,984 __

9,765,625 ≈ 80

23. The series is defined. S = 2 _ 1 − 0.8

25. The series is defined. S = −1 __________ 1 −  − 1 __ 2 

27.

250 500 750

0 x

y

1 2 3 4 5 6 7 8 9 10 11 12

1000 1250 1500 1750 2000

Su m

o f D

ep os

its

Month

29. Sample answer: The graph of Sn seems to be approaching 1. This makes

sense because ∑ k = 1

 1 _ 2  k is a

defined infinite geometric

series with S = 1 __ 2 _________

1 −  1 __ 2  = 1.

31. 49 33. 254 35. S 7 = 147 _ 2 37. S 11 =

55 _ 2

39. S 7 = 5208.4 41. S 10 = − 1023 _ 256

43. S = − 4 _ 3

45. S = 9.2 47. \$3,705.42 49. \$695,823.97 51. ak = 30 − k 53. 9 terms 55. r =

4 _ 5 57. \$400 per month 59. 420 feet 61. 12 feet

Section 13.5 1. There are m + n ways for either event A or event B to occur. 3. The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem. 5. A combination;

C(n, r) = n! _ (n − r)!r!

7. 4 + 2 = 6 9. 5 + 4 + 7 = 16

11. 2 × 6 = 12 13. 103 = 1,000 15. P(5, 2) = 20 17. P(3, 3) = 6 19. P(11, 5) = 55,440 21. C(12, 4) = 495 23. C(7, 6) = 7 25. 210 = 1,024 27. 212 = 4,096

29. 29 = 512 31. 8! _ 3!

= 6,720 33. 12! _ 3!2!3!4!

35. 9

37. Yes, for the trivial cases r = 0 and r = 1. If r = 0, then C(n, r) = P(n, r) = 1. If r = 1, then r = 1, C(n, r) = P(n, r) = n.

39. 6! ___ 2!

× 4! = 8,640 41. 6 − 3 + 8 − 3 = 8 43. 4 × 2 × 5 = 40

45. 4 × 12 × 3 = 144 47. P(15, 9) = 1,816,214,400 49. C(10, 3) × C(6, 5) × C(5, 2) = 7,200 51. 211 = 2,048

53. 20! ______ 6!6!8!

= 116,396,280

Section 13.6 1. A binomial coefficient is an alternative way of denoting the

combination C(n, r). It is defined as  n r  = C(n, r) = n! _________

r !(n − r)! .

3. The Binomial Theorem is defined as (x + y)n = ∑ k = 0

n

 n k  x n − ky k and can be used to expand any binomial. 5. 15 7. 35 9. 10 11. 12,376 13. 64a3 − 48a2b + 12ab2 − b3 15. 27a3 + 54a2b + 36ab2 + 8b3

17. 1024×5 + 2560x4y + 2560x3y2 + 1280x2y3 + 320xy4 + 32y5

19. 1024×5 − 3840x4y + 5760x3y2 − 4320x2y3 + 1620xy4 − 243y5

21. 1 ___ x4

+ 8 ___ x3y

+ 24 ____ x2y2

+ 32 ___ xy3

+ 16 ___ y4

23. a17 + 17a16b + 136a15b2

25. a15 − 30a14b + 420a13b2

27. 3,486,784,401a20 + 23,245,229,340a19b + 73,609,892,910a18b2

29. x 24 − 8x 21 √ — y + 28x 18y 31. −720x 2y 3

33. 220,812,466,875,000y 7 35. 35x 3y 4

37. 1,082,565a 3b 16 39. 1152y2

_ x7

41. f2(x) = x 4 + 12x 3

1 2 3

x

f2(x) y

−2 −1−1 1 2−2−3−4−5

−4 −5

−3

4 5 6 7

45. 590,625x 5y 2

47. k − 1

43. f4(x) = x 4 + 12x 3 + 54x 2 + 108x

h

2

x

y

–1–2–3–4–5–6–7–8–9 –1 –2 –3 –4

1

0 3

3

21 4

4 5 6 7 8 9

10

49. The expression (x3 + 2y2 − z)5 cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

Section 13.7 1. Probability; the probability of an event is restricted to values between 0 and 1, inclusive of 0 and 1. 3. An experiment is an activity with an observable result. 5. The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets A and B and a union of events A and B, the union includes either A or B or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is

always a numerical value between 0 and 1. 7. 1 __ 2

9. 5 __ 8

11. 5 __ 8 13. 3 __ 8

15. 1 __ 4

17. 3 __ 4 19. 3 __ 8

21. 1 __ 8

23. 15 ___ 16

25. 5 __ 8

27. 1 ___ 13

29. 1 ___ 26

31. 12 ___ 13

33.

1 2 3 4 5 6

1 (1, 1) 2 (1, 2)

3 (1, 3)

4 (1, 4)

5 (1, 5)

6 (1, 6)

7

2 (2, 1) 3 (2, 2)

4 (2, 3)

5 (2, 4)

6 (2, 5)

7 (2, 6)

8

3 (3, 1) 4 (3, 2)

5 (3, 3)

6 (3, 4)

7 (3, 5)

8 (3, 6)

9

4 (4, 1) 5 (4, 2)

6 (4, 3)

7 (4, 4)

8 (4, 5)

9 (4, 6)

10

5 (5, 1) 6 (5, 2)

7 (5, 3)

8 (5, 4)

9 (5, 5)

10 (5, 6)

11

6 (6, 1) 7 (6, 2)

8 (6, 3)

9 (6, 4)

10 (6, 5)

11 (6, 6)

12

35. 5 ___ 12

37. 0. 39. 4 __ 9

41. 1 __ 4

43. 3 __ 4

45. 21 ___ 26 47. C(12, 5) ________ C(48, 5)

= 1 _____ 2162

49. C(12, 3)C(36, 2) ______________ C(48, 5)

= 175 _____ 2162

51. C(20, 3)C(60, 17) _______________ C(80, 20)

≈ 12.49% 53. C(20, 5)C(60, 15) _______________ C(80, 20)

≈ 23.33%

55. 20.50 + 23.33 − 12.49 = 31.34%

57. C(40000000, 1)C(277000000, 4) ___________________________ C(317000000, 5)

= 36.78%

59. C(40000000, 4)C(277000000, 1) ___________________________ C(317000000, 5)

= 0.11%

Chapter 13 Review exercises 1. 2, 4, 7, 11 3. 13, 103, 1003, 10003 5. The sequence is arithmetic. The common difference is d = 5 _ 3 . 7. 18, 10, 2, −6, −14 9. a 1 = −20, an = an − 1 + 10

11. an = 1 _ 3 n +

13 _ 24 13. r = 2 15. 4, 16, 64, 256, 1024 17. 3, 12, 48, 192, 768 19. an = −

1 _ 5 ∙  1 _ 3 

n − 1

21. ∑ m = 0

5

 1 __ 2 m + 5  23. S 11 = 110 25. S9 ≈ 23.95

27. S = 135 ____ 4

29. \$5,617.61 31. 6 33. 104 = 10,000

35. P(18, 4) = 73,440 37. C(15, 6) = 5,005

39. 250 = 1.13 × 1015 41. 8! ____ 3!2!

= 3,360 43. 490,314

45. 131,072a 17 + 1,114,112a 16b + 4,456,448a 15b 2

47.   1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

49. 1 __

6 51. 5 __

9 53. 4 __

9 55. 1 − C(350, 8) ________

C(500, 8 ) ≈ 94.4%

57. C(150, 3)C(350, 5) ________________ C(500, 8)

≈ 25.6%

Chapter 13 practice test 1. −14, −6, −2, 0 3. The sequence is arithmetic. The common difference is d = 0.9. 5. a 1 = −2, an = an − 1 −

3 _ 2 ;

a 22 = − 67 _ 2 7. The sequence is geometric. The common ratio

is r = 1 _ 2 . 9. a 1 = 1, an = − 1 _ 2 ∙ an −1 11. ∑ k = −3

15

 3k 2 − 5 __ 6 k  13. S 7 = −2,604.2 15. Total in account: \$140,355.75; Interest earned: \$14,355.75 17. 5 × 3 × 2 × 3 × 2 = 180

19. C(15, 3) = 455 21. 10! _ 2!3!2!

= 151,200 23. 429x 14 _

16

25. 4 _ 7 27. 5 _ 7 29.

C(14, 3)C(26, 4) ______________ C(40, 7)

≈ 29.2%

• Chapter 11. Systems of Equations and Inequalities
• Chapter 11. Systems of Equations and Inequalities
• 11.1. Systems of Linear Equations: Two Variables
• 11.2. Systems of Linear Equations: Three Variables
• 11.6. Solving Systems with Gaussian Elimination
• Glossary
• Key Equations
• Key Concepts
• Review Exercises
• Practice Test
• Chapter 13. Sequences, Probability and Counting Theory
• Chapter 13. Sequences, Probability and Counting Theory
• 13.1. Sequences and their Notations
• 13.2. Arithmetic Sequences
• 13.3. Geometric Sequences
• 13.4. Series and their Notations
• 13.6. Binomial Theorem
• Glossary
• Key Equations
• Key Concepts
• Review Exercises
• Practice Test