13.1.12: Chapter 12

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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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Be Prepared

Try It

Section 12.1 Exercises

Section 12.2 Exercises

Section 12.3 Exercises

Section 12.4 Exercises

Review Exercises

Practice Test