13.1.12: Chapter 12

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Feb 19, 2024
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- 13.1.11: Chapter 11
- Back Matter

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Be Prepared

12.1

5, 7, 9, 11

12.2

$−1, 1, −1, 1$

12.3

12.4

3, 7, 11, 15

12.5

$(5, 2)$

12.6

654

12.7

$\frac{3}{4}$

12.8

ⓐ 81; ⓑ $\frac{1}{16}$

12.9

ⓐ 12; ⓑ 36; ⓒ 108

12.10

12.11

$9 x^{2} + 30 x + 25$

12.12

$x^{2} - 2 x y + y^{2}$

Try It

12.1

$−1, 2, 5, 8, 11$

12.2

$−3, −1, 1, 3, 5$

12.3

$7, 13, 31, 85, 247$

12.4

$−3, −1, 3, 11, 27$

12.5

$−1, 4, −9, 16, −25$

12.6

$1, −8, 27, −64, 125$

12.7

$a_{n} = 3 n$

12.8

$a_{n} = 5 n$

12.9

$a_{n} = {(−1)}^{n} 3^{n}$

12.10

$a_{n} = {(−1)}^{n + 1} n^{2}$

12.11

$a_{n} = \frac{1}{2^{n}}$

12.12

$a_{n} = \frac{1}{n^{2}}$

12.13

$2, 1, \frac{1}{3}, \frac{1}{12}, \frac{1}{60}$

12.14

$3, \frac{3}{2}, \frac{1}{2}, \frac{1}{8}, \frac{1}{40}$

12.15

$\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30}$

12.16

$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}$

12.17

12.18

12.19

$\frac{16}{3}$

12.20

12.21

$\sum_{n = 1}^{5} \frac{1}{2^{n}}$

12.22

$\sum_{n = 1}^{5} \frac{1}{n^{2}}$

12.23

$\sum_{n = 1}^{5} {(−1)}^{n + 1} n^{2}$

12.24

$\sum_{n = 1}^{5} {(−1)}^{n} 2 n$

12.25

ⓐ The sequence is arithmetic with common difference $d = 11$ . ⓑ The sequence is arithmetic with common difference $d = −6$ .
ⓒ The sequence is not arithmetic as all the differences between the consecutive terms are not the same.

12.26

ⓐ The sequence is not arithmetic as all the differences between the consecutive terms are not the same. ⓑ The sequence is arithmetic with common difference $d = 2 .$
ⓒ The sequence is arithmetic with common difference $d = −5 .$

12.27

$7, 3, −1, −5, −9, …$

12.28

$11, 3, −5, −13, −21, …$

12.29

241

12.30

$−106$

12.31

$a_{11} = 2 .$ The general term is $a_{n} = −3 n + 35 .$

12.32

$a_{19} = −55 .$ The general term is $a_{n} = −4 n + 21 .$

12.33

$a_{1} = 5,$ $d = 4 .$ The general term is $a_{n} = 4 n + 1 .$

12.34

$a_{1} = 8,$ $d = −3 .$ The general term is $a_{n} = −3 n + 11 .$

12.35

1,890

12.36

1,515

12.37

2,300

12.38

5,250

12.39

2,670

12.40

3,045

12.41

ⓐ The sequence is geometric with common ratio $r = 3 .$ ⓑ The sequence is geometric with common ratio $r = \frac{1}{4} .$ ⓒ The sequence is not geometric. There is no common ratio.

12.42

ⓐ The sequence is not geometric. There is no common ratio. ⓑ The sequence is geometric with common ratio $r = 2 .$ ⓒ The sequence is geometric with common ratio $r = \frac{1}{2} .$

12.43

$7, −21, 63, −189, 567$

12.44

$6, −24, 96, −384, 1536$

12.45

$\frac{1}{6,561}$

12.46

$\frac{1}{16,384}$

12.47

$a_{9} = 39,366 .$ The general term is $a_{n} = 6 {(3)}^{n - 1} .$

12.48

$a_{11} = 7,168 .$ The general term is $a_{n} = 7 {(2)}^{n - 1} .$

12.49

$3,145,725$

12.50

$10,460,353,200$

12.51

$393,204$

12.52

$10,230$

12.53

12.54

$\frac{256}{3}$

12.55

$\frac{4}{9}$

12.56

$\frac{8}{9}$

12.57

$$ 10,000$

12.58

$$ 3, 333.33$

12.59

$$ 88,868.36$

12.60

$$ 698,201.57$

12.61

$x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3}$
$+ 5 x y^{4} + y^{5}$

12.62

$p^{7} + 7 p^{6} q + 21 p^{5} q^{2} + 35 p^{4} q^{3}$
$+ 35 p^{3} q^{4} + 21 p^{2} q^{5} + 7 p q^{6} + q^{7}$

12.63

$x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16$

12.64

$x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2}$
$+ 6 x + 1$

12.65

$16 x^{4} - 96 x^{3} + 216 x^{2} - 216 x + 81$

12.66

$64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3}$
$+ 60 x^{2} - 12 x + 1$

12.67

ⓐ 6 ⓑ 1 ⓒ 1 ⓓ 35

12.68

ⓐ 2 ⓑ 1 ⓒ 1 ⓓ 6

12.69

$x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3}$
$+ 5 x y^{4} + y^{5}$

12.70

$m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3}$
$+ 15 m^{2} n^{4} + 6 m n^{5} + n^{6}$

12.71

$x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2}$
$+ 405 x - 243$

12.72

$y^{6} - 6 y^{5} + 15 y^{4} - 20 y^{3} + 15 y^{2}$
$- 6 y + 1$

12.73

$243 x^{5} - 810 x^{4} y + 1080 x^{3} y^{2}$
$- 720 x^{2} y^{3} + 240 x y^{4} - 32 y^{5}$

12.74

$256 x^{4} - 768 x^{3} y + 864 x^{2} y^{2}$
$- 432 x y^{3} + 81 y^{4}$

12.75

$15 x^{4} y^{2}$

12.76

$70 a^{4} b^{4}$

12.77

3,584

12.78

280

Section 12.1 Exercises

$−5, −3, −1, 1, 3$

$4, 7, 10, 13, 16$

$5, 7, 11, 19, 35$

$1, 5, 21, 73, 233$

$2, 1, \frac{8}{9}, 1, \frac{32}{25}$

11.

$1, \frac{3}{2}, \frac{5}{4}, \frac{7}{8}, \frac{9}{16}$

13.

$−2, 4, −6, 8, −10$

15.

$1, −4, 9, −16, 25$

17.

$1, - \frac{1}{4}, \frac{1}{9}, - \frac{1}{16}, \frac{1}{25}$

19.

$a_{n} = 8 n$

21.

$a_{n} = n + 5$

23.

$a_{n} = e^{n + 2}$

25.

$a_{n} = {(−1)}^{n} 5 n$

27.

$a_{n} = {(−1)}^{n} n^{3}$

29.

$a_{n} = {(−1)}^{n} 2 n$

31.

$a_{n} = \frac{1}{4^{n}}$

33.

$a_{n} = - \frac{n}{n + 1}$

35.

$a_{n} = - \frac{5}{2^{n}}$

37.

$4, 2, \frac{2}{3}, \frac{1}{6}, \frac{1}{30}$

39.

$3, 6, 18, 72, 360$

41.

$2, 24, 720, 40320, 3628800$

43.

$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$

45.

$1, \frac{1}{2}, \frac{2}{3}, \frac{3}{2}, \frac{24}{5}$

47.

$2, \frac{3}{2}, \frac{8}{3}, \frac{15}{2}, \frac{144}{5}$

49.

$1 + 4 + 9 + 16 + 25 = 55$

51.

$5 + 7 + 9 + 11 + 13 + 15 = 60$

53.

$2 + 4 + 8 + 16 = 30$

55.

$\frac{4}{1} + \frac{4}{1} + \frac{4}{2} + \frac{4}{6} = \frac{32}{3} = 10 \frac{2}{3}$

57.

$2 + 6 + 12 + 20 + 30 = 70$

59.

$\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6} = \frac{71}{20}$

61.

$\sum_{n = 1}^{5} \frac{1}{3^{n}}$

63.

$\sum_{n = 1}^{5} \frac{1}{n^{3}}$

65.

$\sum_{n = 1}^{5} \frac{2}{n}$

67.

$\sum_{n = 1}^{5} {(−1)}^{n + 1} 3 n$

69.

$\sum_{n = 1}^{10} {(−1)}^{n} 2 n$

71.

$\sum_{n = 1}^{7} (2 n + 12)$

73.

Answers will vary.

75.

Answers will vary.

Section 12.2 Exercises

77.

The sequence is arithmetic with common difference $d = 8 .$

79.

The sequence is not arithmetic.

81.

The sequence is arithmetic with common difference $d = −3 .$

83.

$11, 18, 25, 32, 39$

85.

$−7, −3, 1, 5, 9$

87.

$14, 5, −4, −13, −22$

89.

$163$

91.

$131$

93.

$−79$

95.

$a_{20} = −34 .$ The general term is $a_{n} = −2 n + 6 .$

97.

$a_{11} = 59 .$ The general term is $a_{n} = 5 n + 4 .$

99.

$a_{8} = −13 .$ The general term is $a_{n} = −5 n + 27 .$

101.

$a_{1} = 11,$ $d = 3 .$ The general term is $a_{n} = 3 n + 8 .$

103.

$a_{1} = 21,$ $d = −8 .$ The general term is $a_{n} = −8 n + 29 .$

105.

$a_{1} = −15,$ $d = 3 .$ The general term is $a_{n} = 3 n - 18 .$

107.

1,635

109.

$−1, 065$

111.

360

113.

6,325

115.

–3,575

117.

6,280

119.

4,125

121.

$−3,850$

123.

Answers will vary.

125.

Answers will vary.

Section 12.3 Exercises

127.

The sequence is geometric with common ratio $r = 4 .$

129.

The sequence is geometric with common ratio $r = \frac{1}{2} .$

131.

The sequence is geometric with a common ratio $r = −2 .$

133.

The sequence is geometric with common ratio $r = \frac{1}{2} .$

135.

The sequence is arithmetic with common difference $d = 5 .$

137.

The sequence is geometric with common ratio $r = \frac{1}{2} .$

139.

$4, 12, 36, 108, 324$

141.

$−4, 8, −16, 32, −64$

143.

$27, 9, 3, 1, \frac{1}{3}$

145.

$472,392$

147.

3,072

149.

$0.0001$

151.

$a_{9} = 2,304 .$ The general term is $a_{n} = 9 {(2)}^{n - 1} .$

153.

$a_{15} = - \frac{2}{19, 683} .$ The general term is $a_{n} = −486 {(- \frac{1}{3})}^{n - 1} .$

155.

$a_{10} = 0.000000001 .$ The general term is $a_{n} = {(0.1)}^{n - 1} .$

157.

$57,395,624$

159.

$−65,538$

161.

$\frac{7,174,453}{59,049} \approx 121.5$

163.

$65,534$

165.

$4088$

167.

$\frac{29,524}{6561} \approx 4.5$

169.

$\frac{3}{2}$

171.

$\frac{9}{2}$

173.

no sum as $r \geq 1$

175.

2,048

177.

$\frac{1}{3}$

179.

$\frac{7}{9}$

181.

$\frac{5}{11}$

183.

185.

$$ 295,581.88$

187.

$$ 14,234.10$

189.

Answers will vary.

191.

Answers will vary.

Section 12.4 Exercises

193.

$a^{8} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3}$
$+ 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6}$
$+ 8 a b^{7} + b^{8}$

195.

$p^{9} + 9 p^{8} q + 36 p^{7} q^{2} + 84 p^{6} q^{3}$
$+ 126 p^{5} q^{4} + 126 p^{4} q^{5} + 84 p^{3} q^{6}$
$+ 36 p^{2} q^{7} + 9 p q^{8} + q^{9}$

197.

$a^{6} - 6 a^{5} b + 15 a^{4} b^{2} - 20 a^{3} b^{3}$
$+ 15 a^{2} b^{4} - 6 a b^{5} + b^{6}$

199.

$x^{3} + 15 x^{2} + 75 x + 125$

201.

$y^{7} + 7 y^{6} + 21 y^{5} + 35 y^{4} + 35 y^{3}$
$+ 21 y^{2} + 7 y + 1$

203.

$z^{6} - 12 z^{5} + 60 z^{4} - 160 z^{3} + 240 z^{2}$
$- 192 z + 64$

205.

$243 x^{5} - 405 x^{4} + 270 x^{3} - 90 x^{2}$
$+ 15 x - 1$

207.

$27 x^{3} - 135 x^{2} + 225 x - 125$

209.

$27 x^{3} + 135 x^{2} y + 225 x y^{2} + 125 y^{3}$

211.

213.

215.

$m^{5} + 5 m^{4} n + 10 m^{3} n^{2} + 10 m^{2} n^{3}$
$+ 5 m n^{4} + n^{5}$

217.

$s^{7} + 7 s^{6} t + 21 s^{5} t^{2} + 35 s^{4} t^{3}$
$+ 35 s^{3} t^{4} + 21 s^{2} t^{5} + 7 s t^{6} + t^{7}$

219.

$y^{4} - 12 y^{3} + 54 y^{2} - 108 y + 81$

221.

$q^{3} - 12 q^{2} + 48 q - 64$

223.

$625 x^{4} - 1000 x^{3} y + 600 x^{2} y^{2}$
$- 160 x y^{3} + 16 y^{4}$

225.

$243 x^{5} + 1620 x^{4} y + 4320 x^{3} y^{2}$
$+ 5760 x^{2} y^{3} + 3840 x y^{4} + 1024 y^{5}$

227.

$126 a^{5} b^{4}$

229.

$462 x^{5} y^{6}$

231.

112

233.

324

235.

30,618

237.

Answers will vary.

239.

Answers will vary.

Review Exercises

241.

$7, 13, 31, 85, 247$

243.

$\frac{3}{4}, \frac{5}{16}, \frac{7}{64}, \frac{9}{256}, \frac{11}{1024}$

245.

$a_{n} = 9 n$

247.

$a_{n} = e^{n - 4}$

249.

$a_{n} = - \frac{n}{n + 2}$

251.

$\frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30}, \frac{1}{42}$

253.

$−3 + (−1) + 1 + 3 + 5$
$+ 7 + 9 = 21$

255.

$4 + 4 + 2 + \frac{2}{3} + \frac{1}{6} = \frac{65}{6}$

257.

$\sum_{n = 1}^{5} {(−1)}^{n} \frac{1}{3^{n}}$

259.

$\sum_{n = 1}^{5} \frac{4}{n}$

261.

The sequence is arithmetic with common difference $d = 6 .$

263.

$5, 8, 11, 14, 17$

265.

$−13, −7, −1, 5, 11$

267.

$−129$

269.

$a_{18} = 103 .$ The general term is $a_{n} = 7 n - 23 .$

271.

$a_{1} = 1,$ $d = 4 .$ The general term is $a_{n} = 4 n - 3 .$

273.

$−1,095$

275.

$585$

277.

$4,850$

279.

$980$

281.

The sequence is not geometric.

283.

The sequence is geometric with common ratio $r = −2 .$

285.

$128, 32, 8, 2, \frac{1}{2}$

287.

$1,536$

289.

$a_{12} = −25, 165, 824 .$ The general term is $a_{n} = 6 {(−4)}^{n - 1} .$

291.

−43,692

293.

$3906.25$

295.

$\frac{189}{8} = 23.625$

297.

$\frac{343}{6} \approx 57.167$

299.

$\frac{4}{11}$

301.

$$ 1,634,421.27$

303.

$x^{4} - 4 x^{3} y + 6 x^{2} y^{2} - 4 x y^{3} + y^{4}$

305.

$32 y^{5} - 240 y^{4} + 720 y^{3} - 1080 y^{2}$
$+ 810 y - 243$

307.

309.

311.

$t^{9} - 9 t^{8} + 36 t^{7} - 84 t^{6} + 126 t^{5}$
$- 126 t^{4} + 84 t^{3} - 36 t^{2} + 9 t - 1$

313.

$256 x^{4} + 768 x^{3} y + 864 x^{2} y^{2}$
$+ 432 x y^{3} + 81 y^{4}$

315.

$84 a^{3} b^{6}$

317.

135

319.

280

Practice Test

321.

$\frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}$

323.

$−4 + 16 - 64 + 256 = 204$

325.

$−13, −10, −7, −4, −1$

327.

$a_{23} = 59 .$ The general term is $a_{n} = 3 n - 10 .$

329.

$1,325$

331.

$3,260$

333.

The sequence is geometric with common ratio $r = \frac{1}{3} .$

335.

$5,242,880$

337.

$797,162$

339.

$\frac{5}{6}$

341.

$$ 1,409,344.19$

343.

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Section 12.1 Exercises

Section 12.2 Exercises

Section 12.3 Exercises

Section 12.4 Exercises

Review Exercises

Practice Test

Be Prepared

Try It

Section 12.1 Exercises

Section 12.2 Exercises

Section 12.3 Exercises

Section 12.4 Exercises

Review Exercises

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