Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12: Sequences; Induction; the Binomial Theorem
2
63. a1
= 6, r = −
3
Since r < 1, the series converges.
a1
6 6 18
S∞
= = = =
1− r ⎛ ⎛ 2 ⎞⎞ ⎛5
⎞ 5
⎜1−⎜− ⎟ ⎜ ⎟
3
⎟
⎝ ⎝ ⎠⎠
⎝3
⎠
1
64. a1
= 4, r =−
2
Since r < 1, the series converges.
a1
4 4 8
S∞
= = = =
1− r ⎛ ⎛ 1 ⎞⎞ ⎛3
⎞ 3
⎜1−⎜− ⎟ ⎜ ⎟
2
⎟
⎝ ⎝ ⎠⎠
⎝2
⎠
65.
66.
∞ k ∞ k−1 ∞ k−1
⎛2⎞ 2 ⎛2⎞ ⎛2⎞
∑3⎜ ⎟ = ∑3⋅ ⋅ ⎜ ⎟ = ∑ 2⎜ ⎟
k= 1 ⎝3⎠ k= 1 3 ⎝3⎠ k=
1 ⎝3⎠
2
a 1 = 2 , r =
3
Since r < 1, the series converges.
a1
2 2
S∞
= = = = 6
1−
r 1−
2 1
3 3
∞ k ∞ k−1 ∞ k−1
⎛3⎞ 3 ⎛3⎞ 3⎛3⎞
∑2⎜ ⎟ = ∑2⋅ ⋅ ⎜ ⎟ = ∑ ⎜ ⎟
k= 1 ⎝4⎠ k= 1 4 ⎝4⎠ k=
12⎝4⎠
3 3
a
1
= , r =
2 4
Since r < 1, the series converges.
3 3
a1
2 2 3
S∞
= = = = ⋅ 4=
6
1−
r 1−
3 1 2 4 4
67. { n + 2 }
d = ( n+ 1+ 2) − ( n+ 2) = n+ 3−n− 2=
1
The difference between consecutive terms is
constant. Therefore the sequence is arithmetic.
50 50 50
S = ( k + 2) = k+
2
50
68. { 2n − 5}
∑ ∑ ∑
k= 1 k= 1 k=
1
50(50 + 1)
= + 2(50) = 1275 + 100 = 1375
2
d = 2( n+ 1) −5 −(2n−5)
= 2n+ 2−5− 2n+ 5=
2
The difference between consecutive terms is
constant. Therefore the sequence is arithmetic.
50
50 50 50
∑ ∑ ∑
S = (2k − 5) = 2 k−
5
k= 1 k= 1 k=
1
⎛50(50 + 1) ⎞
= 2⎜
⎟− 5(50) = 2550 − 250 = 2300
⎝ 2 ⎠
2
69. { 4n } Examine the terms of the sequence: 4,
16, 36, 64, 100, ...
There is no common difference and there is no
common ratio. Therefore the sequence is neither
arithmetic nor geometric.
2
70. { 5 1}
71.
72.
n + Examine the terms of the sequence:
6, 21, 46, 81, 126, ...
There is no common difference and there is no
common ratio. Therefore the sequence is neither
arithmetic nor geometric.
⎧ 2 ⎫
⎨ 3 − n ⎬
⎩ 3 ⎭
⎛ 2 ⎞ ⎛ 2 ⎞
d = ⎜3 − ( n+ 1) ⎟−⎜3−
n⎟
⎝ 3 ⎠ ⎝ 3 ⎠
2 2 2 2
= 3− n− − 3+ n =−
3 3 3 3
The difference between consecutive terms is
constant. Therefore the sequence is arithmetic.
50 50 50
⎛ 2 ⎞ 2
S50
= ∑⎜
3− k ⎟= ∑3−
∑k
k= 1⎝
3 ⎠ k= 1 3 k=
1
2 ⎛50(50 + 1) ⎞
= 3(50) − ⎜ ⎟= 150 − 850 =−700
3⎝
2 ⎠
⎧ 3 ⎫
⎨ 8 − n ⎬
⎩ 4 ⎭
⎛ 3 ⎞ ⎛ 3 ⎞
d = ⎜8 − ( n+ 1) ⎟−⎜8−
n⎟
⎝ 4 ⎠ ⎝ 4 ⎠
3 3 3 3
= 8− n− − 8+ n =−
4 4 4 4
The difference between consecutive terms is
constant. Therefore the sequence is arithmetic.
50 50 50
S ⎛ 3 3
50
8 8
1 4 k ⎞
= ∑⎜
− ⎟= ∑ − ∑
k k 1 4
k
= ⎝ ⎠ = k=
1
3 ⎛50(50 + 1) ⎞
= 8(50) − ⎜ ⎟
4⎝
2 ⎠
= 400 − 956.25 =−556.25
1258
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