Precalculus with Limits, Answers to Section 9.1 1

Precalculus with Limits, Answers to Section 9.1 1C

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Chapter 9Section 9.1 (page 649)

Vocabulary Check (page 649)1. infinite sequence 2. terms 3. finite4. recursively 5. factorial6. summation notation 7. index; upper; lower8. series 9. th partial sum

1. 4, 7, 10, 13, 16 2. 2, 7, 12, 17, 22 3. 2, 4, 8, 16, 32

4. 5.

6. 7. 8.

9. 10. 11.

12. 0, 2, 0, 2, 0 13. 14.

15. 16.

17. 18.

19. 20. 0.3, 0.3, 0.3, 0.3, 0.3

21. 0, 0, 6, 24, 60 22.

23. 24. 25. 26.

27. 28.

29. 30.

31. 32.

33. c 34. b 35. d 36. a 37.

38. 39.

40. 41.

42. 43. 44.

45. 46. 47.

48. 49. 50.

51. 28, 24, 20, 16, 12 52. 15, 18, 21, 24, 27

53. 3, 4, 6, 10, 18 54. 32, 16, 8, 4, 2

55. 6, 8, 10, 12, 14 56. 25, 20, 15, 10, 5

57. 81, 27, 9, 3, 1 58.

59. 60.

61. 62.

63.

64.

65. 66. 67. 90 68. 600 69.

70. 71. 72.

73. 35 74. 57 75. 40 76. 25 77. 30

78. 110 79. 80. 81. 88 82. 238

83. 30 84. 11 85. 81 86. 6.06 87.

88. 89. 90. 91.

92. 93. 94.

95. 96. 97.

98. 99. 100. 101.

102. 103. 104. 105. 106.

107. (a)

(b)

108. (a)

(b)(c) A240 � $99,914.79

A60 � $8248.64A6 � $621.35A5 � $515.20,A4 � $410.10,A3 � $306.04,A2 � $203.01,A1 � $101.00,

A40 � $11,040.20A8 � $5858.30A7 � $5743.43,

A6 � $5630.81,A5 � $5520.40,A4 � $5412.16,A3 � $5306.04,A2 � $5202.00,A1 � $5100.00,

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Precalculus with Limits, Answers to Section 9.1 2

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d.

(Continued)

109. (a)(b)(c)

The quadratic model is a better fit.(d) The quadratic model; 995

110. (a)

(b) The number of cases reported fluctuates.

111. (a)

(b) The federal debt is increasing.

112. $17,495.1 million

113. True by the Properties of Sums

114. True. The sums are equal because

115. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 1, 2,

116. Answers will vary. 117. $500.95 118. $1.943

119–120. Answers will vary.

121. 122.

123.

124.

125. 126.

127. 128. No inverse

129. (a) (b)

(c) (d)

130. (a) (b)

(c) (d)

131. (a) (b)

(c) (d)

132. (a) (b)

(c) (d)

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05 14

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n 8 9 10 11 12 13

311 357 419 481 548 608

303 363 424 484 545 605

308 362 420 480 544 611cn

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333202CB09_AN.qxd 4/13/06 5:20 PM Page 2


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Section 9.2 (page 659)

Vocabulary Check (page 659)1. arithmetic; common 2.3. sum of a finite arithmetic sequence

1. Arithmetic sequence,


3. Not an arithmetic sequence








11. 8, 11, 14, 17, 20Arithmetic sequence,


13. 7, 3, Arithmetic sequence,


15. 1, 1, Not an arithmetic sequence

16. 1, 2, 4, 8, 16Not an arithmetic sequence

17.Not an arithmetic sequence

18. 2, 8, 24, 64, 160Not an arithmetic sequence

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 5, 11, 17, 23, 29 32.

33.

34. 16.5, 16.75, 17, 17.25, 17.5

35. 2, 6, 10, 14, 18 36. 1, 6, 11, 16, 21

37. 38. 22.45, 20.725, 19, 17.275, 15.55

39. 15, 19, 23, 27, 31;

40. 6, 11, 16, 21, 26;

41. 200, 190, 180, 170, 160;

42. 72, 66, 60, 54, 48;

43.

44. 0.375, 0.625, 0.875, 1.125, 1.375;

45. 59 46. 83 47. 18.6 48. 49. b

50. d 51. c 52. a

53. 54.

55. 56.

57. 620 58. 1850 59. 17.4 60. 23 61. 265

62. 375 63. 4000 64. 16,100 65. 10,000

66. 1220 67. 1275 68. 10,100 69. 30,030

70. 26,425 71. 355 72. 2500 73. 160,000

74. 218,625 75. 520 76. 44,625 77. 2725

78. 79. 10,120 80. 1402.5

81. (a) $40,000 (b) $217,500

82. (a) $45,550 (b) $247,050

83. 2340 seats 84. 2430 seats 85. 405 bricks

86. 203 bricks 87. 490 meters 88. 784 feet

89. (a) (b) $900

90. (a) (b) $7800

91. $70,500; answers will vary.

92. $375,000

93. (a)

(b) $110

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d.

(Continued)

94. (a)

(b) $525.00

95. (a)(b) the models are similar.(c) (d) 2004: $32,960

2005: $34,058

(e) Answers will vary.

96. (a)

(b)

(c) $38,856 (d) $19,366.31

97. True. Given and and

98. True by the formula for the sum of a finite arithmeticsequence,

99. Answers will vary.

100. Add the first term to times the common difference.

101. (a) (b)

(c) The graph of contains all points on theline. The graph of contains only pointsat the positive integers.

(d) The slope of the line and the common difference ofthe arithmetic sequence are equal.

102. (a) 4, 9, 16, 25, 36 (b)

(c)

103. 4

104. answers will vary.

105. Slope: 106. Slope: intercept: intercept:

107. Slope: undefined; 108. Slope: 0No intercept intercept:

109.

110.


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333202CB09_AN.qxd 4/13/06 5:20 PM Page 4


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Vocabulary Check (page 669)1. geometric; common 2.

3. 4. geometric series

5.

1. Geometric sequence,


3. Not a geometric sequence







10. Not a geometric sequence 11. 2, 6, 18, 54, 162

12. 6, 12, 24, 48, 96 13.

14. 15.

16. 17.

18. 19.

20.

21. 64, 32, 16, 8, 4;

22. 81, 27, 9, 3, 1;

23. 7, 14, 28, 56, 112;

24.

25.

26.

27. 28.

29.

30. 31.

32.

33.

34. 35. 45,927

36. 8,957,952 37. 50,388,480 38. 8,388,608

39. 40. 41.

42. 43. a 44. c 45. b 46. d

47. 48.

49. 50.

51. 52.

53. 511 54. 6357.162 55. 171 56.

57. 43 58. 2.667 59. 60.

61. 29,921.31 62. 12.500 63. 592.647 64.

65. 2092.596 66. 3949.147 67. 68. 45.000

69. 6.400 70. 5.333 71. 3.750 72. 45

73. 74. 75.

76. 77. 78.

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97. Horizontal asymptote:Corresponds to the sum of theseries

98. Horizontal asymptote: Corresponds to the sum of theseries

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d.

(Continued)

99. (a)(b) The population is growing at a rate of 0.6% per year.(c) 1342.2 million. This value is close to the prediction.(d) 2007

100. (a) $1790.85 (b) $1806.11 (c) $1814.02

(d) $1819.40 (e) $1822.03

101. (a) $3714.87 (b) $3722.16 (c) $3725.85(d) $3728.32 (e) $3729.52

102. $22,689.45 103. $7011.89

104. $3698.34 105–106. Answers will vary.

107. (a) $26,198.27 (b) $26,263.88

108. (a) $33,534.21 (b) $33,551.91

109. (a) $118,590.12 (b) $118,788.73

110. (a) $76,122.54 (b) $76,533.16

111. Answers will vary. 112. $222,289.91

113. $1600 114. $1250 115.

116. $2000 117. 126 square inches

118.

119. $3,623,993.23

120. (a) 152.42 feet (b) 19 seconds

121. False. A sequence is geometric if the ratios of consecutiveterms are the same.

122. False.

123. Given a real number between and 1, as the exponentincreases, approaches zero.

124. Answers will vary. 125. 126.

127. 128.

129. 130. Does not factor

131. 132.

133. 134.

135. 136.

137. 138.


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Vocabulary Check (page 681)1. mathematical induction 2. first3. arithmetic 4. second

1. 2. 3.

4. 5–34. Answers will vary.

35. 36.

37. 38.

39. 40. 41. 120

42. 465 43. 91 44. 3025 45. 979 46. 61,776

47. 70 48. 43,890 49. 50. 195

51. 0, 3, 6, 9, 12, 15First differences: 3, 3, 3, 3, 3Second differences: 0, 0, 0, 0Linear

52. 2, 4, 6, 8, 10, 12First differences: 2, 2, 2, 2, 2Second differences: 0, 0, 0, 0Linear

53. 3, 1, First differences: Second differences: Quadratic

54.First differences:Second differences:Neither

55. 2, 4, 16, 256, 65,536, 4,294,967,296First differences: 2, 12, 240, 65,280, 4,294,901,760Second differences: 10, 228, 65,040, 4,294,836,480Neither

56. 0, 1, 3, 6, 10, 15First differences: 1, 2, 3, 4, 5Second differences: 1, 1, 1, 1Quadratic

57. 58.

59. 60.

61. (a) 2.2, 2.4, 2.2, 2.3, 0.9(b) A linear model can be used.

(c)(d) Part b: Part c:

These are very similar.62. Answers will vary. 63. True. may be false.64. False. must be proven to be true.65. True. If the second differences are all zero, then the first

differences are all the same and the sequence is arithmetic.66. False. A sequence with terms has second

differences.67. 68.69.70.71. (a) Domain: all real numbers except

(b) Intercept:(c) Vertical asymptote:

Horizontal asymptote:(d)

72. (a) Domain: all real numbers except (b) Intercept:(c) Vertical asymptotes:

Horizontal asymptote:(d)

73. (a) Domain: all real numbers except (b) -intercept:(c) Vertical asymptote:

Horizontal asymptote: (d) y

t−8 −6 −4 −2

−8

−6

−4

2

4

62 8

(7, 0)

y � 1t � 0

�7, 0�tt � 0t

y

x−8 −6 −4

2

4

6

8

4 6 8

(0, 0)

y � 1x � ±2

�0, 0�x � ±2x

y

x−12−10 −8 −6 −4

−6

−4

2

4

6

8

10

2 4

(0, 0)

y � 1x � �3

�0, 0�x � �3x

8x3 � 48x2y � 96xy2 � 64y3

�64x3 � 240x2 � 300x � 1254x2 � 4xy � y24x4 � 4x2 � 1

n � 2n

P1

P7

an � 141.34an � 142.3;an � 2.08n � 103.9

an � 2.2n � 102.7

an �74n2 � 5n � 3an �

12 n 2 � n � 3

an � n2 � 2n � 7an � n2 � n � 3

�27, 54, �108, 2169, �18, 36, �72, 144

�3, 6, �12, 24, �48, 96

�1, �1, �1, �1�2, �3, �4, �5, �6

�2, �6, �11, �17

�3402

Sn �n

2�n � 2�Sn �n

2�n � 1�

Sn �65 �1 � ��3

2�n�Sn � 10 � 10� 910�n

Sn �n2

��3n � 53�Sn � n�2n � 1�

k � 13

�2k � 3�

�k � 1�2�k � 2�2

41

2�k � 3�5

�k � 1��k � 2�

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(Continued)

74. (a) Domain: all real numbers except

(b) intercept: intercept:

(c) Vertical asymptotes:

Horizontal asymptote:

(d) y

x

−2

−4

−6

−8

4

6

8

2 4 6 8(−5, 0)

(0, 5)

y � �1

x � 1

�0, 5�y-��5, 0�;x-

x � 1x

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Vocabulary Check (page 688)1. binomial coefficients2. Binomial Theorem; Pascal’s Triangle

3. 4. expanding a binomial

1. 10 2. 28 3. 1 4. 1 5. 15,504

6. 792 7. 210 8. 210 9. 4950 10. 4950

11. 56 12. 8 13. 35 14. 20

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39. 40. 41.

42. 43.

44. 45.

46. 47. 1,732,104

48. 3,247,695 49. 180 50. 720

51. 52. 16,128 53. 210 54. 45

55.

56.

57.

58.

59.

60.

61.

62.

63. 64. 65.

66. 67. 1 68.

69. 1.172 70. 1049.890 71. 510,568.785

72. 467.721

73. is shifted four units to the left of

74.

is shifted three units to the right of

75. 0.273 76. 0.250 77. 0.171 78. 0.273

79. (a)(b)

(c)(d)

00 13

60

f

g

g�t� � 0.0025t 3 � 0.06t 2 � 1.33t � 17.5

00 13

24

f �t� � 0.0025t 3 � 0.015t 2 � 0.88t � 7.7

g�x� � �x4 � 12x3 � 50x2 � 84x � 46f.g

g

−4

−3

8

f

5

g�x� � x3 � 12x2 � 44x � 48f.

g

−8

−4

4

fg

4

184 � 4403 i�10 � 198i

2035 � 828i�38 � 41i�4

h � 0�1

x�x � h�,

h � 01

x � h � x,

h � 04x3 � 6x 2h � 4xh 2 � h3,

h � 03x 2 � 3xh � h 2,

u3 � 10u125 � 40u95 � 80u65 � 80u35 � 32

x 2 � 3x 43y 13 � 3x 23y 23 � y

8t32 � 12t � 6t12 � 1

x2 � 12x32 � 54x � 108x12 � 81

�326,592

1.293 � 1013x9y6

32,476,950,000x4y832,400ab4

1,259,712x2y7�35,000x4z3

360x3y2y6120x7y3

� 2916v � 729

64v6 � 576v5 � 2160v 4 � 4320v3 � 4860v2

x5 � 10x4y � 40x3y2 � 80x2y3 � 80xy4 � 32y5

81 � 216z � 216z2 � 96z3 � 16z4

32t 5 � 80t 4s � 80t 3s2 � 40t 2s3 � 10ts4 � s5

3x5 � 15x 4 � 26x3 � 18x 2 � 3x � 1

2x4 � 24x3 � 113x2 � 246x � 207

1

x6�

12y

x 5�

60y2

x4�

160y3

x3�

240y4

x2�

192y5

x� 64y6

1

x 5�

5y

x 4�

10y2

x 3�

10y3

x 2�

5y4

x� y5

� 6x 2y 10 � y12

x12 � 6x10y 2 � 15x8y4 � 20x 6y 6 � 15x 4y8

x8 � 4x6y2 � 6x 4y 4 � 4x2y6 � y8

343a3 � 147a2b � 21ab2 � b3

8x3 � 12x2y � 6xy2 � y3

� 6250xy 4 � 3125y5

32x5 � 400x 4y � 2000x3y2 � 5000x2y3

� 3840ab4 � 1024b5

243a5 � 1620a4b � 4320a3b2 � 5760a2b3

x4 � 8x3y � 24x2y 2 � 32xy3 � 16y4

� 1458rs5 � 729s6

r 6 � 18r 5s � 135r 4s2 � 540r 3s3 � 1215r 2s4

c3 � 3c2d � 3cd 2 � d3

x5 � 5x 4y � 10x 3y2 � 10x 2y 3 � 5xy 4 � y5

y5 � 10y 4 � 40y3 � 80y 2 � 80y � 32

y3 � 12y2 � 48y � 64

a5 � 25a4 � 250a3 � 1250a2 � 3125a � 3125

a4 � 24a3 � 216a2 � 864a � 1296

x 6 � 6x 5 � 15x4 � 20x 3 � 15x 2 � 6x � 1

x 4 � 4x3 � 6x 2 � 4x � 1

�nr�; nCr

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(Continued)

(e) 33.26 gallons; 33.26 gallons; yes(f) The trend is for the per capita consumption of bottled

water to increase. This may be due to the increasingconcern with contaminants in tap water.

80. (a)(b)

(c)

81. True. The coefficients from the Binomial Theorem can beused to find the numbers in Pascal’s Triangle.

82. False. Expanding binomials that represent differences isaccurate. The coefficients have alternating signs.

83. False. The coefficient of the -term is 1,732,104 and thecoefficient of the -term is 192,456.

84. The first and last numbers in each row are 1. Every othernumber in each row is formed by adding the two numbersimmediately above the number.

85.

86. terms

87. The signs of the terms in the expansion of alternatebetween positive and negative.

88.

is the expansion of 89–92. Answers will vary.93. 94.

95. 96.

97. 98. �2010

11.56��4

5�5�6�

g�x� � �x � 1 � 2g�x� � x � 2 � 1

x1 5

3

2

1

2 3 4−1−2−3

−4

−5

y

x1

3

5

4

−3 2 3−1−2−1

1

2

y

g�x� � ��x � 2�2 � 3g�x� � �x � 3�2

x4 53

5

1−1−2−3−5

3

−2

4

2

−3

−4

−5

y

x4 6

4

8

2

6

2

−2

−2−4

y

f �x�.k �x�

−4

−6 6

4

h

g

p k = f

�x � y�n

n � 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 1045 1

x14x10

f �t�: 2007, g�t�: 2007

00 20

60

f

g

g�t� � 0.031t 2 � 1.44t � 17.4

g�t�:f �t�:

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Vocabulary Check (page 698)1. Fundamental Counting Principle 2. permutation

3. 4. distinguishable permutations

5. combinations

1. 6 2. 6 3. 5 4. 3 5. 3 6. 4

7. 8 8. 6 9. 30 10. 15 11. 30 12. 1440

13. 64 14. 4096 15. 175,760,000 16. 5,760,000

17. (a) 900 (b) 648 (c) 180 (d) 600

18. (a) 9000 (b) 4536 (c) 4000 (d) 4500

19. 64,000 20. 125,000 21. (a) 40,320 (b) 384

22. (a) 40,320 (b) 576 23. 24 24. 120

25. 336 26. 380 27. 120 28. 840

29. or 30. or

31. 1,860,480 32. 9,034,502,400 33. 970,200

34. 1,814,400 35. 15,504 36. 120 37. 120

38. 720 39. 11,880 40. 24 41. 420

42. 56 43. 2520 44. 34,650

45. ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD,BADC, CABD, CADB, DABC, DACB, BCAD, BDAC,CBAD, CDAB, DBAC, DCAB, BCDA, BDCA, CBDA,CDBA, DBCA, DCBA

46. ABCD, DBCA, ACBD, DCBA 47. 1,816,214,400

48. 120 49. 5,586,853,480 50.

51. AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE,DF, EF

52. 15,504 53. 324,632 54. 3,838,380

55. (a) 35 (b) 63 (c) 203

56. (a) 3 (b) 28 (c) 66 (d) 190

57. (a) 3744 (b) 24 58. (a) 70 (b) 30

59. 292,600 60. (a) (b) 691,530

61. 5 62. 9 63. 20 64. 35

65. (a) 146,107,962

(b) If the jackpot is won, there is only one winning number.

(c) There are 28,989,675 possible winning numbers in the state lottery, which is considerably less than thepossible number of winning Powerball numbers.

66. (a) Permutation because order matters.

(b) Combination because order does not matter.

(c) Permutation because order matters.

(d) Combination because order does not matter.

67. False. It is an example of a combination.

68. True by the definition of the Fundamental CountingPrinciple.

69. They are equal.

70. Changing the order of any of the six elementsselected results in a different permutation but the samecombination.

71–74. Proof

75. No. For some calculators the number is too great.

76. The symbol denotes the number of ways to choose andorder elements out of a collection of elements.

77. (a) 35 (b) 8 (c) 83

78. (a) 2 (b) 4 (c)

79. (a) (b) 0 (c) 0

80. (a) 29 (b) (c) 445 81. 8.30

82. 5.5 83. 35 84. 8.32

�3

�4

x � 2 � 2

nrnPr

10P6 > 10C6.

1.335�10�10

4.42 � 1016

n � 10n � 9n � 6n � 5

nPr �n!

�n � r�!

333202CB09_AN.qxd 4/13/06 5:20 PM Page 11


Vocabulary Check (page 709)1. experiment; outcomes 2. sample space3. probability 4. impossible; certain5. mutually exclusive 6. independent7. complement 8. (a) iii (b) i (c) iv (d) ii

1.

2.

3.

4.

5.

6. 7.

8. 9. 10. 11. 12. 13.

14. (for A–6) 15. 16. 17. 18.

19. 20. 21. 22. 23. 24.

25. 0.3 26. 0.64 27. 28. 29. 0.86

30. 0.08 31. 32.

33. (a) 58% (b) 95.6% (c) 0.4%

34. (a) 34% (b) 45% (c) 23%

35. (a) 243 (b) (c)

36. (a) (b) (c)

37. (a) (b) (c)

38. (a) (b) (c)

39.

40. 19% 41. (a) (b) (c)

42. (a) (b) 43. (a) (b)

44. (a) 0.076 (b) 0.00069 45. (a) (b) (c)

46. 47. (a) (b) (c)

48. (a) (b) (c) (d) 49. 0.4746

50. 0.1024 51. (a) 0.9702 (b) 0.9998 (c) 0.0002

52. (a) 0.81 (b) 0.01 (c) 0.99

53. (a) (b) (c) 54.

55. (a) (b) (c) (d) (e)

(f) The probabilities are slightly better in European roulette.

56. (a) (b) Approximations will vary.

57. True. Two events are independent if the occurrence of onehas no effect on the occurrence of the other.

58. False. The complement of the event is to roll a numbergreater than or equal to 3, and its probability is

59. (a) As you consider successive people with distinct birth-days, the probabilities must decrease to take into accountthe birth dates already used. Because the birth dates ofpeople are independent events, multiply the respectiveprobabilities of distinct birthdays.

(b) (c) Answers will vary.(d) is the probability that the birthdays are not distinct,

which is equivalent to at least two people having thesame birthday.

(e)

(f) 23

60. Meteorological records indicate that over an extended period of time with similar weather conditions it will rain40% of the time.

61. No real solution 62.

63. 64. 65. 66.

67. 68. 69. 70. 3

71. 72.

73. 74.y

x−4 −3

−4

−3

1

3

4

1 3 4

y

x−8 −6 −4 −2

−14

−12

−8

2

4 6 8

y

x−8 −6 −4 −2

−4

−6

−8

4

2

4 6 8

y

x−4 −2 2

12

10

8

4

2

864 12

�10�1112

±4�40, ±10, 1 ± 13

2

�3 ± 574

Qn

365365 � 364

365 � 363365 � 362

365

23.

4

7296859

11444

1019

919

138

716

1516

18

116

140

8411600

12

14

5455

1255

1455

64165

413

12

513

124

1120

1930

38

49323

225646

211292

P��Moore wins�� P��Jenkins wins�� 14

P��Taylor wins�� 12

12101

39101

62101

274627

97209

112209

87100

17100

59100

1625

150

39100

1835

13

34

1115

25

115

15

1936

13

19

1112

712

112

613

326

1013

313

12

78

12

38�SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF�

BD, BE, CD, CE, DE��AB, AC, AD, AE, BC,

�blue, blue�, �blue, yellow��red, red�, �red, blue�, �red, yellow�,

�ABC, ACB, BAC, BCA, CAB, CBA��2, 3, . . . , 12��T, 1�, �T, 2�, �T, 3�, �T, 4�, �T, 5�, �T, 6��H, 1�, �H, 2�, �H, 3�, �H, 4�, �H, 5�, �H, 6�,


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n 10 15 20 23 30 40 50

0.88 0.75 0.59 0.49 0.29 0.11 0.03

0.12 0.25 0.41 0.51 0.71 0.89 0.97Qn

Pn

333202CB09_AN.qxd 4/13/06 5:20 PM Page 12

Precalculus with Limits, Answers to Review Exercises 13

Review Exercises (page 715)1. 8, 5, 4, 2.

3. 72, 36, 12, 3, 4. 0, 2, 6, 12, 20 5.

6. 7. 8.

9. 120 10. 12 11. 1 12. 13. 30 14. 56

15. 16. 6.17 17. 6050 18. 35 19.

20. 21. 22. 23. 24.

25. (a)

(b)

26. (a)




30. Arithmetic sequence, 31. 4, 7, 10, 13, 16

32. 6, 4, 2, 0, 33. 25, 28, 31, 34, 37

34. 4.2, 4.6, 5.0, 5.4, 5.8 35.

36. 37.

38. 39.

40. 41. 80 42. 52 43. 88

44. 250 45. 25,250 46. 3050

47. (a) $43,000 (b) $192,500 48. 676 bales




52. Not a geometric sequence 53.

54. 2, 4, 8, 16, 32 55. 9, 6, 4, or

56. or

57.

58.

59.

60. 61. 127 62. 121

63. 64. 65. 31 66. 720 67. 24.85

68. 25 69. 5486.45 70. 1493.50 71. 8

72. 73. 74. 2 75. 12 76.

77. (a) (b) $20,168.40

78. $32,939.75; $32,967.03 79–82. Answers will vary.

83. 84.

85. 86.

87. 465 88. 385 89. 4648 90. 12,110

91. 5, 10, 15, 20, 25

First differences: 5, 5, 5, 5

Second differences: 0, 0, 0

Linear

92.

First differences:

Second differences:

Quadratic

93. 16, 15, 14, 13, 12

First differences:

Second differences: 0, 0, 0

Linear

94. 0, 1, 1, 2, 2

First differences: 1, 0, 1, 0

Second differences:

Neither

95. 15 96. 120 97. 56 98. 220

99. 35 100. 126 101. 28 102. 10

103.

104.

105.

106.

107. 108. 109. 11

110. 180 111. 10,000 112. 72 113. 720

114. 225,792,840 115. 56 116. 327,680 117.

118. 119. (a) 43% (b) 82%

120. (a) 41.6% (b) 80% (c) 7.4% 121.

122. 123. 124.

125. True.

126. True by Properties of Sums

127. True by Properties of Sums

128. True. The sums are equal because

23�2 � 24�2 � 25�2 � 26�2 � 27�2 � 28�2.

21 � 22 � 23 � 24 � 25 � 26 �

�n � 2�!n!

��n � 2��n � 1�n!

n!� �n � 2��n � 1�

3132

34

5324

1216

1120

19

�236 � 115i41 � 840i

� 189x 2y 10 � 21xy 12 � y 14

2187x 7 � 5103x 6y 2 � 5103x 5y 4 � 2835x 4y 6 � 945x 3y 8

a5 � 15a 4b � 90a3b2 � 270a2b3 � 405ab4 � 243b5

x6 � 18x5 � 135x4 � 540x3 � 1215x2 � 1458x � 729

x 4 � 16x3 � 96x2 � 256x � 256

�1, 1, �1

�1, �1, �1, �1

�2, �2, �2

�4, �6, �8, �10

�3, �7, �13, �21, �31

Sn �14413 �1 � �� 1

12�n�Sn �52�1 � �3

5�n�Sn � 4n�18 � n�Sn � n�2n � 7�

at � 120,000�0.7�t

139

109

32

364243

1516

an � 5�0.2�n�1; 2.621 � 10�13

an � 100�1.05�n�1; 252.695

an � 1296�16�n

; 3.545 � 10�13

an � 16��12�n�1

; 3.052 � 10�5

2, �26, 12, �126, 722, 26, 12, 126, 72

9, �6, 4, �83, 16

983, 16

9

4, �1, 14, � 116, 1

64

r � �2

r � �13

r � 2

an � �13 n �

313

an � �7n � 107an � nx � 3x

an � 3ny � 2yan � �3n � 28

an � 12n � 5

�2

d � �19

d �12

d � �20

4 13

1000

a12 � 920.28a11 � 889.57,a10 � 861,a9 � 834.57,a8 � 810.28,

a7 � 788.13,a6 � 768.12,a5 � 750.25,a4 � 734.52,

A120 � $22,196.40A10 � $10,687.03A9 � $10,616.25,A8 � $10,545.95,A7 � $10,476.10,A6 � $10,406.73,A5 � $10,337.81,A4 � $10,269.35,A3 � $10,201.34,A2 � $10,133.78,A1 � $10,066.67,

110

299

1

359�

9

k�1

kk � 1

�20

k�1

12k

20524

18

an ��1�n�1

nan �

4n

an � n2 � 2

an � 2��1�n35

�5, 103 , �3, 20

7 , �259

72, 16

5

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333202CB09_AN.qxd 4/13/06 5:20 PM Page 13

(Continued)

129. False. When equals 0 or 1, then the results are the same.

130. The set of natural numbers

131. In the sequence in part (a), the odd-numbered terms arenegative, whereas in the sequence in part (b), the even-numbered terms are negative.

132. (a) Arithmetic. There is a constant difference betweenconsecutive terms.

(b) Geometric. Each term is a constant multiple of thepreceding term. In this case, the common ratio isgreater than 1.

133. Each term of the sequence is defined in terms of preced-ing terms.

134. Increasing powers of real numbers between 0 and 1approach zero.

135. d 136. a 137. b 138. c

139. 240, 440, 810, 1490, 2740

140. Closed interval0 ≤ p ≤ 1;

r

Precalculus with Limits, Answers to Review Exercises 14

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Precalculus with Limits, Answers to Chapter Test 15

Chapter Test (page 719)

1.

2.

3. 50, 61, 72; 140

4.

5. 5, 10, 20, 40, 80

6. 86,100

7. 189

8. 4


10.

11.

12. (a) 72 (b) 328,440

13. (a) 330 (b) 720,720

14. 26,000

15. 720 16.

17.

18. 25%

3.908 � 10�10

115

�108,864

x4 � 8x3y � 24x2y2 � 32xy3 � 16y4

an � 0.8n � 1.4

an �n � 2

n!

�15

, 18

, �1

11,

114

, �1

17

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Cumulative Test for Chapters 7–9 (page 720)1. 2.

3. 4.

5. 6.

7.

Maximum at

Minimum at

8. $0.75 mixture: 120 pounds; $1.25 mixture: 80 pounds

9.

10. 11.

12. 13. 14.

15. 16. 84 17.

18. Gym shoes: $198.36 millionJogging shoes: $358.48 millionWalking shoes: $167.17 million

19. 20. 21. 9

22. 23.

24. 920 25. (a) 65.4 (b)

26. 3, 6, 12, 24, 48 27. 28. Answers will vary.

29. 30. 210

31. 60032. 70 33. 120 34. 453,600

35. 151,200 36. 720 37. 14

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8

10

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Precalculus with Limits, Answers to Cumulative Test for Chapters 7–9 16

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333202CB09_AN.qxd 4/13/06 5:20 PM Page 16

Precalculus with Limits, Answers to Problem Solving 17

Problem Solving (page 725)

1.

approaches

2. (a) (b) 0

(c)

(d) 0

3. (a)

(b) If is odd, and if is even, (c)

(d) It is not possible to find the value of as approachesinfinity.

4. (a) Arithmetic sequence, difference(b) Arithmetic sequence, difference(c) Not an arithmetic sequence

5. (a) 3, 5, 7, 9, 11, 13, 15, 17

(b) To obtain the arithmetic sequence, find the differencesof consecutive terms of the sequence of perfect cubes.Then find the differences of consecutive terms of thissequence.

(c) 12, 18, 24, 30, 36, 42, 48

(d) To obtain the arithmetic sequence, find the thirdsequence obtained by taking differences of consecutiveterms in consecutive sequences.

(e) 60, 84, 108, 132, 156, 180

6.

This represents the total distance Achilles ran.

This represents the total amount of time Achilles ran.

7.

8. (a) 7, 22, 11, 34, 17, 52, 26, 13, 40, 20(b) Eventually the terms repeat: 4, 2, 1 if is a positive

integer and if is a negative integer.


10. (a) is true for integers (b) is true for integers (c) are true.(d) is true for any positive integer

11. (a) Answers will vary. (b) 17,710

12. (a) 30 marbles (b) 3 to 7; 7 to 3

(c)

(d) Odds in favor of event

13. 14.

15. (a) (b) 2.53, 24 turns�$0.71

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13

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n ≥ 3.Pn

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an � 6n � 6

an � 2n � 1

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00 10

8

00 10

2

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1.414213562, 1.414213562, . . . 1, 1.5, 1.416, 1.414215686,

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333202CB09_AN.qxd 4/13/06 5:20 PM Page 17

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Precalculus with Limits, Answers to Section 9.1 1