Chapter 12 Sequences; Induction; the Binomial Theorem

Section 12.5: The Binomial Theorem
29. n = 10, j = 4, x = x, a = 3
⎛10⎞ 10! 10987 ⋅ ⋅ ⋅
⎜ ⎟x ⋅ 3 = ⋅ 81x = ⋅81x
⎝ 4 ⎠ 4!6! 4⋅3⋅2⋅1
6
= 17,010x
6
The coefficient of x is 17,010.
6 4 6 6
30. n= 10, j = 7, x = x, a = − 3
⎛10⎞
3 7 10!
3
⎜ ⎟x
⋅− ( 3) = ⋅− ( 2187)
x
⎝ 7 ⎠ 7!3!
10⋅9⋅8
= ⋅( −2187)
x
321 ⋅ ⋅
3
=−262,440x
3
The coefficient of x is − 262,440.
31. n = 12, j = 5, x = 2 x, a = − 1
⎛12⎞
7 5 12! 7
⎜ ⎟(2 x) ⋅− ( 1) = ⋅128 x ( −1)
⎝ 5 ⎠
5!7!
12⋅11⋅10⋅9⋅8 = ⋅ ( − 128) x
54321 ⋅ ⋅ ⋅ ⋅
7
=−101,376x
7
The coefficient of x is − 101,376.
32. n= 12, j = 9, x = 2 x, a = 1
⎛12⎞ 12! 12⋅11⋅10
⎜ ⎟(2 x) ⋅ (1) = ⋅ 8 x (1) = ⋅8x
⎝ 9 ⎠
9!3! 3⋅2⋅1
3
= 1760x
3
The coefficient of x is 1760.
3 9 3 3
33. n= 9, j = 2, x = 2 x, a = 3
⎛9⎞
7 2 9! 7
⎜ ⎟(2 x) ⋅ 3 = ⋅128 x (9)
⎝2⎠
2!7!
98 ⋅ 128
7
= ⋅ x ⋅ 9
21 ⋅
7
= 41,472x
7
The coefficient of x is 41,472.
34. n= 9, j = 7, x = 2 x, a = − 3
⎛9⎞
2 7 9! 2
⎜ ⎟(2 x) ⋅− ( 3) = ⋅4 x ( −2187)
⎝7⎠
7!2!
98 ⋅ 4
2
= ⋅ x ⋅− 2187
21 ⋅
2
=−314,928x
2
The coefficient of x is − 314,928.
3
7
35. n= 7, j = 4, x = x, a = 3
⎛7⎞
7! 7⋅6⋅5
⎜ ⎟x ⋅ 3 = ⋅ 81x = ⋅ 81x = 2835x
⎝4⎠
4!3! 3⋅2⋅1
3 4 3 3 3
36. n= 7, j = 2, x = x, a = − 3
⎛7⎞
7! 7⋅6
⎜ ⎟x ⋅− ( 3) = ⋅ 9x = ⋅ 9x = 189x
⎝2⎠
2!5! 2⋅1
5 2 5 5 5
37. n= 9, j = 2, x = 3 x, a =− 2
⎛9⎞
7 2 9!
7
⎜ ⎟(3 x) ⋅− ( 2) = ⋅2187x
⋅4
⎝2⎠
2!7!
98 ⋅
= ⋅ 8748 x = 314,928 x
21 ⋅
38. n= 8, j = 5, x = 3 x, a = 2
7 7
⎛8⎞
3 5 8! 3
⎜ ⎟(3 x) ⋅ ( 2) = ⋅27x
⋅32
⎝5⎠
5!3!
876 ⋅ ⋅
= ⋅ 864 x = 48,384 x
321 ⋅ ⋅
0
39. The x term in
∑
12 12
⎛12⎞ 12−
j ⎛1 ⎞ 12
( )
j ⎛ ⎞
⎜ ⎟ x
x
j= 0
j
⎜ ⎟ = ∑ ⎜ ⎟
⎝ ⎠ ⎝ x ⎠ j=
0⎝ j ⎠
occurs when:
24 − 3 j = 0
24 = 3 j
j = 8
3 3
2 24−3j
⎛12⎞ 12! 12⋅11⋅10 ⋅9
The coefficient is ⎜ 495
8 ⎟ = = =
⎝ ⎠ 8!4! 4321 ⋅ ⋅ ⋅
0
40. The x term in
∑
9 j 9
⎛9⎞ − ⎛ 1 ⎞ ⎛9⎞
⎜ ⎟ x
1
2
j= 0
j
⎜− ⎟ = ∑ ⎜ ⎟ −
⎝ ⎠ ⎝ x ⎠ j=
0⎝ j⎠
occurs when:
9− 3j
= 0
9=
3j
j = 3
The coefficient is
⎛9⎞
( ) 3 9! 9⋅8⋅7
⎜ ⎟ − 1 =− =− =−84
⎝3⎠
3!6! 3⋅2⋅1
( ) ( )
9 j j 9−3j
x
1271
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