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Precalculus with Limits, Answers to Section 9.1 1C
opyr
ight
©H
ough
ton
Mif
flin
Com
pany
. All
righ
ts r
eser
ved.
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,
29
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19
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n � 11336
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, �1
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, �1
362,880
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, 1
24,
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, 1
40,320
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24, 1120
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8
an � 14��2�n�1an �
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n � 2an � ��1�n�1�2n�
an � n 2 � 1an � 4n � 1
an � 3n � 2
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10
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23, 23, 23, 23, 23
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333202CB09_AN.qxd 4/13/06 5:20 PM Page 1
Precalculus with Limits, Answers to Section 9.1 2
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
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)
133. 26 134. 135. 136. �11,758�194�126
�2021
415
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242
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5, x ≥ 0
g�1�x� �3x
f �1�x� �x � 3
4
�x3
6,
x5
120, �
x7
5040,
x9
362,880, �
x11
39,916,800
�x2
2,
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24, �
x6
720,
x8
40,320, �
x10
3,628,800
�x3
3,
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5, �
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7,
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9, �
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x2
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x3
6,,
x4
24,
x5
120
8955
5534,34
21,2113,13
8 ,85,5
3,32,
21 � 22 � 23 � 24 � 23�2 � 24�2 � 25�2 � 26�2.
00 14
7000
a12 � $6251.5, a13 � $6616.3a9 � $5550.9, a10 � $5735.5, a11 � $5963.5,a6 � $5091.8, a7 � $5245.7, a8 � $5393.2,a3 � $4425.3, a4 � $4698.2, a5 � $4914.8,a0 � $3102.9, a1 � $3644.3, a2 � $4079.6,
05 14
75
a13 � 44.7a9 � 44.8, a10 � 41.4, a11 � 40.1, a12 � 41.1,a5 � 73.1, a6 � 64.3, a7 � 56.5, a8 � 50.0,
cn � 1.61n2 � 26.8n � 9.5bn � 60.57n � 182
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
bn
an
333202CB09_AN.qxd 4/13/06 5:20 PM Page 2
Precalculus with Limits, Answers to Section 9.2 3C
opyr
ight
©H
ough
ton
Mif
flin
Com
pany
. All
righ
ts r
eser
ved.
Section 9.2 (page 659)
Vocabulary Check (page 659)1. arithmetic; common 2.3. sum of a finite arithmetic sequence
1. Arithmetic sequence,
2. Arithmetic sequence,
3. Not an arithmetic sequence
4. Not an arithmetic sequence
5. Arithmetic sequence,
6. Arithmetic sequence,
7. Not an arithmetic sequence
8. Arithmetic sequence,
9. Not an arithmetic sequence
10. Not an arithmetic sequence
11. 8, 11, 14, 17, 20Arithmetic sequence,
12. 97, 94, 91, 88, 85Arithmetic sequence,
13. 7, 3, Arithmetic sequence,
14. 1, 5, 9, 13, 17Arithmetic 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
an � 1300 � 100n
an � �25n � 225
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an � �10n � 210d � �10;
an � 5n � 1d � 5;
an � 4n � 11d � 4;
�2, 2, 6, 10, 14
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5, 174 , 72, 11
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an � �15n � 265an � �3n � 103
an � 5n � 9an �103 n �
53
an � �5n � 15an � �52n �
132
an � 5yn � 6yan � 2xn � x
an � �23n �
23an � �8n � 108
an � 4n � 11an � 3n � 2
�35
34,�1,3
2,�3,
�1�1,�1,
d � 4
d � �4�9�5,�1,
d � �3
d � 3
d � 0.4
d � �12
d � �14
d � 3
d � �2
an � dn � c
Month 1 2 3 4 5 6
Monthly$220 $218 $216 $214 $212 $210payment
Unpaid$1800 $1600 $1400 $1200 $1000 $800balance
333202CB09_AN.qxd 4/13/06 5:20 PM Page 3
Precalculus with Limits, Answers to Section 9.2 4
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
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.
111. Answers will vary.
x � 2, y � �6, z � 3
x � 1, y � 5, z � �1
y
x−8 −6 −4 −2
−12
−10
−8
−6
−4
−2
2
4
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y
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−4
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2
4
6
8
8 10 12 14
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y
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−12
−10
−8
−6
2
4
1 2 3 4
y
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−4
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1
2
3
4
1 2 3 4
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�9;12;
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n
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n 2; 49 � 72
an � 2 � 3ny � 3x � 2
y
x
3330272421181512963
1 2 3 4 5 6 7 8 9 10 11
an
n
3330272421181512963
1 2 3 4 5 6 7 8 9 10 11−1
�n � 1�
Sn �n2
�a1 � an�.
a1 � �n � 1�d.an �d � a2 � a1a2,a1
�13
n�7 an;
an � 1726.93n � 11,718.43
2,000
4,000
6,000
8,000
n
an
10,000
12,000
Rev
enue
(in
mill
ions
of
dolla
rs)
Year (7 ↔ 1997)
7 8 9 10 11 12 13
20,0003 13
32,000
an � 1114.9n � 17,795;an � 1098n � 17,588
Month 5 6 7 8
Payment $290.00 $287.50 $285.00 $282.50
Unpaid$3750 $3500 $3250 $3000balance
Month 9 10 11 12
Payment $280.00 $277.50 $275.00 $272.50
Unpaid$2750 $2500 $2250 $2000balance
Month 1 2 3 4
Payment $300.00 $297.50 $295.00 $292.50
Unpaid$4750 $4500 $4250 $4000balance
333202CB09_AN.qxd 4/13/06 5:20 PM Page 4
Precalculus with Limits, Answers to Section 9.3 5C
opyr
ight
©H
ough
ton
Mif
flin
Com
pany
. All
righ
ts r
eser
ved.
Section 9.3 (page 669)
Vocabulary Check (page 669)1. geometric; common 2.
3. 4. geometric series
5.
1. Geometric sequence,
2. Geometric sequence,
3. Not a geometric sequence
4. Not a geometric sequence
5. Geometric sequence,
6. Geometric sequence,
7. Geometric sequence,
8. Geometric sequence,
9. 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.
79. 2 80. 6 81. �2
3� 82. �
5
6� 83. �
1
3
6�
84. �1
9
0� 85. �
3
5� 86. 5 87. 88. ��
2
2
5�
89. 32 90. 27 91. Undefined 92. Undefined
93. �1
4
1� 94. �
3
1
7
1� 95. �
2
7
2� 96. �
1
2
8
5�
97. Horizontal asymptote:Corresponds to the sum of theseries
98. Horizontal asymptote: Corresponds to the sum of theseries
y � 10
−9
−25
18
20
y � 12
−4
−15
10
20
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n�132�3
4�n�1�6
n�10.1�4�n�1�
6
n�115��1
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00 10
400
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10
16
a7 �256243
a6 � �2a1 � 12a3 � 9
an � 1000�1.005�n�1; 1342.139
an � 500�1.02�n�1; 1082.372
an � �3�n�1; 273
an � 100ex�n�1�; 100e8xan � 64��14�n�1
; � 14096
an � 6��1
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; �2
310
an � 5�32�
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; 10,935
128an � 4�1
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128
an � �96��12�n
r � �12;48, �24, 12, �6, 3;
an � �4��32�n
6, �9, 272 , �81
4 , 2438 ; r � �
32;
an � �52��2�nr � �2;5, �10, 20, �40, 80;
an �72�2�nr � 2;
an � 243�13�nr �
13;
an � 128� 12�n
r �12;
5, 10x, 20x2, 40x3, 80x4
2, x2
, x2
8,
x3
32,
x4
1283, 35, 15, 155, 75
1, e, e2, e3, e46, �32, 38, � 3
32, 3128
5, �12, 1
20, � 1200, 1
20001, 13, 19, 127, 1
81
1, 12, 14, 18, 116
r � �23
r � 2
r � 0.2
r � �12
r � 4
r � 3
S �a1
1 � r
Sn � a1�1 � r n
1 � r �an � a1r
n�1
13. 1, �1
2�, �
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4�, �
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2
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8
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333202CB09_AN.qxd 4/13/06 5:20 PM Page 5
Precalculus with Limits, Answers to Section 9.3 6
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
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.
139. Answers will vary.
7x2 � 21x � 53�x � 1��x � 4�
5x2 � 9x � 30�x � 2��x � 2�
1, x � 3, 52x � 1
3, x � 0, �
12
13
, x � 2, �73x
x � 3, x � �3
4x2�2 � x��2 � x��3x � 1��2x � 5�
x�3x � 8��3x � 8�
9x2 � 24x � 153x2 � 6x � 1
3x � 4x2 � 2x
r nn�1r
an � a1rn�1
S � $2653.80 million
$2181.82
an � 1190.88�1.006�n
333202CB09_AN.qxd 4/13/06 5:20 PM Page 6
Precalculus with Limits, Answers to Section 9.4 7C
opyr
ight
©H
ough
ton
Mif
flin
Com
pany
. All
righ
ts r
eser
ved.
Section 9.4 (page 681)
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�
333202CB09_AN.qxd 4/13/06 5:20 PM Page 7
Precalculus with Limits, Answers to Section 9.4 8
Cop
yrig
ht ©
Hou
ghto
n M
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in C
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ny. A
ll ri
ghts
res
erve
d.
(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
333202CB09_AN.qxd 4/13/06 5:20 PM Page 8
Precalculus with Limits, Answers to Section 9.5 9C
opyr
ight
©H
ough
ton
Mif
flin
Com
pany
. All
righ
ts r
eser
ved.
Section 9.5 (page 688)
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
333202CB09_AN.qxd 4/13/06 5:20 PM Page 9
Precalculus with Limits, Answers to Section 9.5 10
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
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ny. A
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ghts
res
erve
d.
(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�:
333202CB09_AN.qxd 4/13/06 5:20 PM Page 10
Precalculus with Limits, Answers to Section 9.6 11C
opyr
ight
©H
ough
ton
Mif
flin
Com
pany
. All
righ
ts r
eser
ved.
Section 9.6 (page 698)
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
Section 9.7 (page 709)
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�,
Precalculus with Limits, Answers to Section 9.7 12
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d.
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)
27. Arithmetic sequence,
28. Not an arithmetic sequence
29. Arithmetic sequence,
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
49. Geometric sequence,
50. Geometric sequence,
51. Geometric sequence,
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
9. Answers will vary.
10.
11.
12. (a) 72 (b) 328,440
13. (a) 330 (b) 720,720
14. 26,000
15. 720 16.
17.
18. 25%
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114
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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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Precalculus with Limits, Answers to Cumulative Test for Chapters 7–9 16
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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.
9. Answers will vary.
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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