Chapter 12 Sequences; Induction; the Binomial Theorem
Chapter 12 Sequences; Induction; the Binomial Theorem
Chapter 12 Sequences; Induction; the Binomial Theorem
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<strong>Chapter</strong> <strong>12</strong>: <strong>Sequences</strong>; <strong>Induction</strong>; <strong>the</strong> <strong>Binomial</strong> <strong>Theorem</strong><br />
58.<br />
59.<br />
60.<br />
⎛4⎞ ⎛4⎞ ⎛4⎞ ⎛4⎞ ⎛4⎞<br />
( x− 3) = ⎜ ⎟x + ⎜ ⎟x ( − 3) + ⎜ ⎟x ( − 3) + ⎜ ⎟x( − 3) + ⎜ ⎟x<br />
( −3)<br />
⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠<br />
4 4 3 2 2 3 0 4<br />
4 3 2<br />
= x + 4( − 3) x + 6⋅ 9x + 4( − 27) x+<br />
81<br />
4 3 2<br />
= x − <strong>12</strong>x + 54x − 108x+<br />
81<br />
⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞<br />
(2x+ 3) = ⎜ ⎟(2 x) + ⎜ ⎟(2 x) ⋅ 3 + ⎜ ⎟(2 x) ⋅ 3 + ⎜ ⎟(2 x) ⋅ 3 + ⎜ ⎟(2 x) ⋅ 3 + ⎜ ⎟⋅3<br />
⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎝5⎠<br />
5 5 4 3 2 2 3 1 4 5<br />
5 4 3 2<br />
= 32x + 5⋅16x ⋅ 3 + 10⋅8x ⋅ 9 + 10⋅4x ⋅ 27 + 5⋅2x⋅ 81+ 1⋅243<br />
5 4 3 2<br />
= 32x + 240x + 720x + 1080x + 810x+<br />
243<br />
⎛4⎞ ⎛4⎞ ⎛4⎞ ⎛4⎞ ⎛4⎞<br />
(3x− 4) = ⎜ ⎟(3 x) + ⎜ ⎟(3 x) ( − 4) + ⎜ ⎟(3 x) ( − 4) + ⎜ ⎟(3 x)( − 4) + ⎜ ⎟( −4)<br />
⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠<br />
4 4 3 2 2 3 4<br />
4 3 2<br />
= 81x + 4⋅27 x ( − 4) + 6⋅9x ⋅ 16 + 4⋅3 x( − 64) + 1⋅256<br />
4 3 2<br />
= 81x − 432x + 864x − 768x+<br />
256<br />
_________________________________________________________________________________________________<br />
61. n = 9, j = 2, x = x, a = 2<br />
⎛9⎞<br />
9! 9⋅8<br />
⎜ ⎟x ⋅ 2 = ⋅ 4x = ⋅ 4x = 144x<br />
⎝2⎠<br />
2!7! 2⋅1<br />
7<br />
The coefficient of x is 144.<br />
7 2 7 7 7<br />
62. n = 8, j = 5, x = x, a =− 3<br />
⎛8⎞<br />
3 5 8!<br />
3<br />
⎜ ⎟x<br />
( − 3) = ( −243)<br />
x<br />
⎝5⎠<br />
5!3!<br />
876 ⋅ ⋅<br />
= ( − 243) x =− 13,608 x<br />
321 ⋅ ⋅<br />
3<br />
The coefficient of x is − 13,608.<br />
63. n = 7, j = 5, x = 2 x, a = 1<br />
3 3<br />
⎛7⎞<br />
7! 7⋅6<br />
⎜ ⎟(2 x) ⋅ 1 = ⋅ 4 x (1) = ⋅ 4x = 84x<br />
⎝5⎠<br />
5!2! 2⋅1<br />
2<br />
The coefficient of x is 84.<br />
2 5 2 2 2<br />
64. n = 8, j = 2, x = 2 x, a = 1<br />
⎛8⎞<br />
6 2 8! 6<br />
⎜ ⎟(2 x) ⋅ 1 = ⋅64 x (1)<br />
⎝2⎠<br />
2!6!<br />
87 ⋅<br />
= ⋅ 64 x = 1792 x<br />
21 ⋅<br />
6<br />
The coefficient of x is 1792.<br />
6 6<br />
65. This is an arithmetic sequence with<br />
a1 = 80, d =− 3, n=<br />
25<br />
a. a 25 = 80 + (25 −1)( − 3) = 80 − 72 = 8 bricks<br />
25<br />
S = (80 + 8) = 25(44) = 1100 bricks<br />
2<br />
1100 bricks are needed to build <strong>the</strong> steps.<br />
b. 25<br />
66. This is an arithmetic sequence with<br />
a1 = 30, d =− 1, a n = 15<br />
15 = 30 + ( n −1)( −1)<br />
− 15 =− n + 1<br />
− 16 =−n<br />
n = 16<br />
16<br />
S 16 = (30 + 15) = 8(45) = 360 tiles<br />
2<br />
360 tiles are required to make <strong>the</strong> trapezoid.<br />
67. This is a geometric sequence with<br />
3<br />
a1<br />
= 20, r = .<br />
4<br />
a. After striking <strong>the</strong> ground <strong>the</strong> third time, <strong>the</strong><br />
3<br />
⎛3⎞ 135<br />
height is 20⎜<br />
⎟ = ≈ 8.44 feet .<br />
⎝4⎠<br />
16<br />
th<br />
b. After striking <strong>the</strong> ground <strong>the</strong> n time, <strong>the</strong><br />
n<br />
⎛3<br />
⎞<br />
height is 20 ⎜ ⎟ feet .<br />
⎝4<br />
⎠<br />
<strong>12</strong>78<br />
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