Intermediate Algebra

# Chapter 12

Intermediate AlgebraChapter 12

### Try It

12.1

$−1 , 2 , 5 , 8 , 11 −1 , 2 , 5 , 8 , 11$

12.2

$−3 , −1 , 1 , 3 , 5 −3 , −1 , 1 , 3 , 5$

12.3

$7 , 13 , 31 , 85 , 247 7 , 13 , 31 , 85 , 247$

12.4

$−3 , −1 , 3 , 11 , 27 −3 , −1 , 3 , 11 , 27$

12.5

$−1 , 4 , −9 , 16 , −25 −1 , 4 , −9 , 16 , −25$

12.6

$1 , −8 , 27 , −64 , 125 1 , −8 , 27 , −64 , 125$

12.7

$a n = 3 n a n = 3 n$

12.8

$a n = 5 n a n = 5 n$

12.9

$a n = ( −1 ) n 3 n a n = ( −1 ) n 3 n$

12.10

$a n = ( −1 ) n + 1 n 2 a n = ( −1 ) n + 1 n 2$

12.11

$a n = 1 2 n a n = 1 2 n$

12.12

$a n = 1 n 2 a n = 1 n 2$

12.13

$2 , 1 , 1 3 , 1 12 , 1 60 2 , 1 , 1 3 , 1 12 , 1 60$

12.14

$3 , 3 2 , 1 2 , 1 8 , 1 40 3 , 3 2 , 1 2 , 1 8 , 1 40$

12.15

$1 2 , 1 6 , 1 12 , 1 20 , 1 30 1 2 , 1 6 , 1 12 , 1 20 , 1 30$

12.16

$1 2 , 1 3 , 1 4 , 1 5 , 1 6 1 2 , 1 3 , 1 4 , 1 5 , 1 6$

12.17

45

12.18

60

12.19

$16 3 16 3$

12.21

$∑ n = 1 5 1 2 n ∑ n = 1 5 1 2 n$

12.22

$∑ n = 1 5 1 n 2 ∑ n = 1 5 1 n 2$

12.23

$∑ n = 1 5 ( −1 ) n + 1 n 2 ∑ n = 1 5 ( −1 ) n + 1 n 2$

12.24

$∑ n = 1 5 ( −1 ) n 2 n ∑ n = 1 5 ( −1 ) n 2 n$

12.25

The sequence is arithmetic with common difference $d=11d=11$. The sequence is arithmetic with common difference $d=−6d=−6$.
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 $d=2.d=2.$
The sequence is arithmetic with common difference $d=−5.d=−5.$

12.27

$7 , 3 , −1 , −5 , −9 , … 7 , 3 , −1 , −5 , −9 , …$

12.28

$11 , 3 , −5 , −13 , −21 , … 11 , 3 , −5 , −13 , −21 , …$

12.29

241

12.30

$−106 −106$

12.31

$a11=2.a11=2.$ The general term is $an=−3n+35.an=−3n+35.$

12.32

$a19=−55.a19=−55.$ The general term is $an=−4n+21.an=−4n+21.$

12.33

$a1=5,a1=5,$$d=4.d=4.$ The general term is $an=4n+1.an=4n+1.$

12.34

$a1=8,a1=8,$$d=−3.d=−3.$ The general term is $an=−3n+11.an=−3n+11.$

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 $r=3.r=3.$ The sequence is geometric with common ratio $d=14.d=14.$ 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 $r=2.r=2.$ The sequence is geometric with common ratio $r=12.r=12.$

12.43

$7 , −21 , 63 , −189 , 567 7 , −21 , 63 , −189 , 567$

12.44

$6 , −24 , 96 , −384 , 1536 6 , −24 , 96 , −384 , 1536$

12.45

$1 6,561 1 6,561$

12.46

$1 16,384 1 16,384$

12.47

$a9=39,366.a9=39,366.$ The general term is $an=6(3)n−1.an=6(3)n−1.$

12.48

$a11=7,168.a11=7,168.$ The general term is $an=7(2)n−1.an=7(2)n−1.$

12.49

$3,145,725 3,145,725$

12.50

$10,460,353,200 10,460,353,200$

12.51

$393,204 393,204$

12.52

$10,230 10,230$

12.53

96

12.54

$256 3 256 3$

12.55

$4 9 4 9$

12.56

$8 9 8 9$

12.57

$10,000 10,000$

12.58

$3 , 333.33 3 , 333.33$

12.59

$88,868.36 88,868.36$

12.60

$698,201.57 698,201.57$

12.61

$x 5 + 5 x 4 y + 10 x 3 y 2 + 10 x 2 y 3 x 5 + 5 x 4 y + 10 x 3 y 2 + 10 x 2 y 3$
$+ 5 x y 4 + y 5 + 5 x y 4 + y 5$

12.62

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

12.63

$x 4 + 8 x 3 + 24 x 2 + 32 x + 16 x 4 + 8 x 3 + 24 x 2 + 32 x + 16$

12.64

$x 6 + 6 x 5 + 15 x 4 + 20 x 3 + 15 x 2 x 6 + 6 x 5 + 15 x 4 + 20 x 3 + 15 x 2$
$+ 6 x + 1 + 6 x + 1$

12.65

$16 x 4 − 96 x 3 + 216 x 2 − 216 x + 81 16 x 4 − 96 x 3 + 216 x 2 − 216 x + 81$

12.66

$64 x 6 − 192 x 5 + 240 x 4 − 160 x 3 64 x 6 − 192 x 5 + 240 x 4 − 160 x 3$
$+ 60 x 2 − 12 x + 1 + 60 x 2 − 12 x + 1$

12.67

6 1 1 35

12.68

2 1 1 6

12.69

$x 5 + 5 x 4 y + 10 x 3 y 2 + 10 x 2 y 3 x 5 + 5 x 4 y + 10 x 3 y 2 + 10 x 2 y 3$
$+ 5 x y 4 + y 5 + 5 x y 4 + y 5$

12.70

$m 6 + 6 m 5 n + 15 m 4 n 2 + 20 m 3 n 3 m 6 + 6 m 5 n + 15 m 4 n 2 + 20 m 3 n 3$
$+ 15 m 2 n 4 + 6 m n 5 + n 6 + 15 m 2 n 4 + 6 m n 5 + n 6$

12.71

$x 5 − 15 x 4 + 90 x 3 − 270 x 2 x 5 − 15 x 4 + 90 x 3 − 270 x 2$
$+ 405 x − 243 + 405 x − 243$

12.72

$y 6 − 6 y 5 + 15 y 4 − 20 y 3 + 15 y 2 y 6 − 6 y 5 + 15 y 4 − 20 y 3 + 15 y 2$
$− 6 y + 1 − 6 y + 1$

12.73

$243 x 5 − 810 x 4 y + 1080 x 3 y 2 243 x 5 − 810 x 4 y + 1080 x 3 y 2$
$− 720 x 2 y 3 + 240 x y 4 − 32 y 5 − 720 x 2 y 3 + 240 x y 4 − 32 y 5$

12.74

$256 x 4 − 768 x 3 y + 864 x 2 y 2 256 x 4 − 768 x 3 y + 864 x 2 y 2$
$− 432 x y 3 + 81 y 4 − 432 x y 3 + 81 y 4$

12.75

$15 x 4 y 2 15 x 4 y 2$

12.76

$8 a b 7 8 a b 7$

12.77

7,168

12.78

280

### Section 12.1 Exercises

1.

$−5 , −3 , −1 , 1 , 3 −5 , −3 , −1 , 1 , 3$

3.

$4 , 7 , 10 , 13 , 16 4 , 7 , 10 , 13 , 16$

5.

$5 , 7 , 11 , 19 , 35 5 , 7 , 11 , 19 , 35$

7.

$1 , 5 , 21 , 73 , 233 1 , 5 , 21 , 73 , 233$

9.

$2 , 1 , 8 9 , 1 , 32 25 2 , 1 , 8 9 , 1 , 32 25$

11.

$1 , 3 2 , 5 4 , 7 8 , 9 16 1 , 3 2 , 5 4 , 7 8 , 9 16$

13.

$−2 , 4 , −6 , 8 , −10 −2 , 4 , −6 , 8 , −10$

15.

$1 , −4 , 9 , −16 , 25 1 , −4 , 9 , −16 , 25$

17.

$1 , − 1 4 , 1 9 , − 1 16 , 1 25 1 , − 1 4 , 1 9 , − 1 16 , 1 25$

19.

$a n = 8 n a n = 8 n$

21.

$a n = n + 5 a n = n + 5$

23.

$a n = e n + 2 a n = e n + 2$

25.

$a n = ( −1 ) n 5 n a n = ( −1 ) n 5 n$

27.

$a n = ( −1 ) n n 3 a n = ( −1 ) n n 3$

29.

$a n = ( −1 ) n 2 n a n = ( −1 ) n 2 n$

31.

$a n = 1 4 n a n = 1 4 n$

33.

$a n = − n n + 1 a n = − n n + 1$

35.

$− 5 2 n − 5 2 n$

37.

$4 , 2 , 2 3 , 1 6 , 1 30 4 , 2 , 2 3 , 1 6 , 1 30$

39.

$3 , 6 , 18 , 72 , 360 3 , 6 , 18 , 72 , 360$

41.

$2 , 24 , 720 , 40320 , 3628800 2 , 24 , 720 , 40320 , 3628800$

43.

$1 , 1 2 , 1 3 , 1 4 , 1 5 1 , 1 2 , 1 3 , 1 4 , 1 5$

45.

$1 , 1 2 , 2 3 , 3 2 , 24 5 1 , 1 2 , 2 3 , 3 2 , 24 5$

47.

$2 , 3 2 , 8 3 , 15 2 , 144 5 2 , 3 2 , 8 3 , 15 2 , 144 5$

49.

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

51.

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

53.

$2 + 4 + 8 + 16 = 30 2 + 4 + 8 + 16 = 30$

55.

$4 1 + 4 1 + 4 2 + 4 6 + 32 3 = 10 2 3 4 1 + 4 1 + 4 2 + 4 6 + 32 3 = 10 2 3$

57.

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

59.

$1 2 + 2 3 + 3 4 + 4 5 + 5 6 = 71 20 1 2 + 2 3 + 3 4 + 4 5 + 5 6 = 71 20$

61.

$∑ n = 1 5 1 3 n ∑ n = 1 5 1 3 n$

63.

$∑ n = 1 5 1 n 3 ∑ n = 1 5 1 n 3$

65.

$∑ n = 1 5 2 n ∑ n = 1 5 2 n$

67.

$∑ n = 1 5 ( −1 ) n + 1 3 n ∑ n = 1 5 ( −1 ) n + 1 3 n$

69.

$∑ n = 1 10 ( −1 ) n 2 n ∑ n = 1 10 ( −1 ) n 2 n$

71.

$∑ n = 1 7 ( 2 n + 12 ) ∑ n = 1 7 ( 2 n + 12 )$

73.

75.

### Section 12.2 Exercises

77.

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

79.

The sequence is not arithmetic.

81.

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

83.

$11 , 18 , 25 , 32 , 39 11 , 18 , 25 , 32 , 39$

85.

$−7 , −3 , 1 , 5 , 9 −7 , −3 , 1 , 5 , 9$

87.

$14 , 5 , −4 , −13 , −22 14 , 5 , −4 , −13 , −22$

89.

$163 163$

91.

$131 131$

93.

$−79 −79$

95.

$a20=−34.a20=−34.$ The general term is $an=−2n+6.an=−2n+6.$

97.

$a11=59.a11=59.$ The general term is $an=5n+4.an=5n+4.$

99.

$a8=−13.a8=−13.$ The general term is $an=−5n+27.an=−5n+27.$

101.

$a1=11,a1=11,$$d=3.d=3.$ The general term is $an=3n+8.an=3n+8.$

103.

$a1=21,a1=21,$$d=−8.d=−8.$ The general term is $an=−8n+29.an=−8n+29.$

105.

$a1=−15,a1=−15,$$d=3.d=3.$ The general term is $an=3n−18.an=3n−18.$

107.

1,635

109.

$−1 , 065 −1 , 065$

111.

360

113.

6,325

115.

–3,575

117.

6,280

119.

4,125

121.

$−3,850 −3,850$

123.

125.

### Section 12.3 Exercises

127.

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

129.

The sequence is geometric with common ratio $r=12.r=12.$

131.

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

133.

The sequence is geometric with common ratio $r=12.r=12.$

135.

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

137.

The sequence is geometric with common ratio $r=12.r=12.$

139.

$4 , 12 , 36 , 108 , 324 4 , 12 , 36 , 108 , 324$

141.

$−4 , 8 , −16 , 32 , −64 −4 , 8 , −16 , 32 , −64$

143.

$27 , 9 , 3 , 1 , 1 3 27 , 9 , 3 , 1 , 1 3$

145.

$472,392 472,392$

147.

3,072

149.

$0.0001 0.0001$

151.

$a9=2,304.a9=2,304.$ The general term is $an=9(2)n−1.an=9(2)n−1.$

153.

$a15=−219,683.a15=−219,683.$ The general term is $an=−486(−13)n−1.an=−486(−13)n−1.$

155.

$a10=0.000000001.a10=0.000000001.$ The general term is $an=(0.1)n−1.an=(0.1)n−1.$

157.

$57,395,624 57,395,624$

159.

$−65,538 −65,538$

161.

$7,174,453 59,049 ≈ 121.5 7,174,453 59,049 ≈ 121.5$

163.

$65,534 65,534$

165.

$4088 4088$

167.

$29,524 6561 ≈ 4.5 29,524 6561 ≈ 4.5$

169.

$3 2 3 2$

171.

$9 2 9 2$

173.

no sum as $r≥1r≥1$

175.

2,048

177.

$1 3 1 3$

179.

$7 9 7 9$

181.

$5 11 5 11$

183.

$6666.676666.67$ $40004000$ $15,00015,000$ $75007500$

185.

$295,581.88 295,581.88$

187.

$14,234.10 14,234.10$

189.

191.

### Section 12.4 Exercises

193.

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

195.

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

197.

$a 6 − 6 a 5 b + 15 a 4 b 2 − 20 a 3 b 3 a 6 − 6 a 5 b + 15 a 4 b 2 − 20 a 3 b 3$
$+ 15 a 2 b 4 − 6 a b 5 + b 6 + 15 a 2 b 4 − 6 a b 5 + b 6$

199.

$x 3 + 15 x 2 + 75 x + 125 x 3 + 15 x 2 + 75 x + 125$

201.

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

203.

$z 6 − 12 z 5 + 60 z 4 − 160 z 3 + 240 z 2 z 6 − 12 z 5 + 60 z 4 − 160 z 3 + 240 z 2$
$− 192 z + 64 − 192 z + 64$

205.

$243 x 5 − 405 x 4 + 270 x 3 − 90 x 2 243 x 5 − 405 x 4 + 270 x 3 − 90 x 2$
$+ 15 x − 1 + 15 x − 1$

207.

$27 x 3 − 135 x 2 + 225 x − 125 27 x 3 − 135 x 2 + 225 x − 125$

209.

$27 x 3 + 135 x 2 y + 225 x y 2 + 125 y 3 27 x 3 + 135 x 2 y + 225 x y 2 + 125 y 3$

211.

7 1 1 45

213.

4 1 1 55

215.

$m 5 + 5 m 4 n + 10 m 3 n 2 + 10 m 2 n 3 m 5 + 5 m 4 n + 10 m 3 n 2 + 10 m 2 n 3$
$+ 5 m n 4 + n 5 + 5 m n 4 + n 5$

217.

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

219.

$y 4 − 12 y 3 + 54 y 2 − 108 y + 81 y 4 − 12 y 3 + 54 y 2 − 108 y + 81$

221.

$q 3 − 12 q 2 + 48 q − 64 q 3 − 12 q 2 + 48 q − 64$

223.

$625 x 4 − 1000 x 3 y + 600 x 2 y 2 625 x 4 − 1000 x 3 y + 600 x 2 y 2$
$− 160 x y 3 + 16 y 4 − 160 x y 3 + 16 y 4$

225.

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

227.

$126 a 5 b 4 126 a 5 b 4$

229.

$462 x 5 y 6 462 x 5 y 6$

231.

112

233.

324

235.

30,618

237.

239.

### Review Exercises

241.

$7 , 13 , 31 , 85 , 247 7 , 13 , 31 , 85 , 247$

243.

$3 4 , 5 16 , 7 64 , 9 256 , 11 1024 3 4 , 5 16 , 7 64 , 9 256 , 11 1024$

245.

$a n = 9 n a n = 9 n$

247.

$a n = e n − 4 a n = e n − 4$

249.

$a n = − n n + 2 a n = − n n + 2$

251.

$1 6 , 1 12 , 1 20 , 1 30 , 1 42 1 6 , 1 12 , 1 20 , 1 30 , 1 42$

253.

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

255.

$4 + 4 + 2 + 2 3 + 1 6 = 65 6 4 + 4 + 2 + 2 3 + 1 6 = 65 6$

257.

$∑ n = 1 5 ( −1 ) n 1 3 n ∑ n = 1 5 ( −1 ) n 1 3 n$

259.

$∑ n = 1 5 4 n ∑ n = 1 5 4 n$

261.

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

263.

$5 , 8 , 11 , 14 , 17 5 , 8 , 11 , 14 , 17$

265.

$−13 , −7 , −1 , 5 , 11 −13 , −7 , −1 , 5 , 11$

267.

$−129 −129$

269.

$a18=103.a18=103.$ The general term is $an=7n−23.an=7n−23.$

271.

$a1=1,a1=1,$$d=4.d=4.$ The general term is $an=4n−3.an=4n−3.$

273.

$−430 −430$

275.

$585 585$

277.

$4850 4850$

279.

$980 980$

281.

The sequence is not geometric.

283.

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

285.

$128 , 32 , 8 , 2 , 1 2 128 , 32 , 8 , 2 , 1 2$

287.

$1,536 1,536$

289.

$a12=−25,165,824.a12=−25,165,824.$ The general term is $an=6(−4)n−1.an=6(−4)n−1.$

291.

5,460

293.

$≈ 3906.25 ≈ 3906.25$

295.

$189 8 = 23.625 189 8 = 23.625$

297.

$343 6 ≈ 57.167 343 6 ≈ 57.167$

299.

$4 11 4 11$

301.

$1,634,421.27 1,634,421.27$

303.

$x 4 − 4 x 3 y + 6 x 2 y 2 − 4 x y 3 + y 4 x 4 − 4 x 3 y + 6 x 2 y 2 − 4 x y 3 + y 4$

305.

$32 y 5 − 240 y 4 + 720 y 3 − 1080 y 2 32 y 5 − 240 y 4 + 720 y 3 − 1080 y 2$
$+ 810 y − 243 + 810 y − 243$

307.

11 1 1 56

309.

1 1 1 55

311.

$t 9 − 9 t 8 + 36 t 7 − 84 t 6 + 126 t 5 t 9 − 9 t 8 + 36 t 7 − 84 t 6 + 126 t 5$
$− 126 t 4 + 84 t 3 − 36 t 2 + 9 t − 1 − 126 t 4 + 84 t 3 − 36 t 2 + 9 t − 1$

313.

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

315.

$84 a 6 b 3 84 a 6 b 3$

317.

135

319.

280

### Practice Test

321.

$1 4 , 1 5 , 1 6 , 1 7 , 1 8 1 4 , 1 5 , 1 6 , 1 7 , 1 8$

323.

$−4 + 16 − 64 + 256 = 204 −4 + 16 − 64 + 256 = 204$

325.

$−13 , −10 , −7 , −4 , −1 −13 , −10 , −7 , −4 , −1$

327.

$a23=59.a23=59.$ The general term is $an=3n−10.an=3n−10.$

329.

$1,325 1,325$

331.

$3,260 3,260$

333.

The sequence is geometric with common ratio $r=13.r=13.$

335.

$5,242,880 5,242,880$

337.

$797,162 797,162$

339.

$5 6 5 6$

341.

$1,409,344.19 1,409,344.19$

343.

8 1 1 210

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