Chapter 12 Sequences; Induction; the Binomial Theorem

Section 12.3: Geometric Sequences; Geometric Series
21.
22.
a
a
a
a
5
n
5
n
5−1 4
⎛1⎞ ⎛1⎞
= 0⋅ ⎜ ⎟ = 0⋅ ⎜ ⎟ = 0
⎝2⎠ ⎝2⎠
n−1
⎛1
⎞
= 0⋅ ⎜ ⎟ = 0
⎝2
⎠
5−1 4
⎛ 1⎞ ⎛ 1⎞
1
= 1⋅⎜− ⎟ = 1⋅⎜− ⎟ =
⎝ 3⎠ ⎝ 3⎠
81
n−1 n−1
⎛ 1⎞ ⎛ 1⎞
= 1⋅⎜− ⎟ = ⎜−
⎟
⎝ 3⎠ ⎝ 3⎠
−
23. a5
( ) ( )
n−1
n
an
= 2⋅ ( 2) = ( 2)
24.
a
a
5
n
5 1 4
= 2⋅ 2 = 2⋅ 2 = 2⋅ 4=
4 2
5−1 4
⎛1⎞ ⎛1⎞
= 0⋅ ⎜ ⎟ = 0⋅ ⎜ ⎟ = 0
⎝π⎠ ⎝π⎠
n−1
⎛1
⎞
= 0⋅ ⎜ ⎟ = 0
⎝π
⎠
1
25. a1
= 1, r = , n = 7
2
a
7
7−1 6
⎛1⎞ ⎛1⎞
1
= 1⋅ ⎜ ⎟ = ⎜ ⎟ =
⎝ 2 ⎠ ⎝ 2 ⎠ 64
26. a1
= 1, r = 3, n = 8
27.
a
8
1
9
8−1 7
= 1⋅ 3 = 3 = 2187
a = 1, r =− 1, n = 9
a
−
( ) ( )
9 1 8
= 1⋅ − 1 = − 1 = 1
10
n
32. a 1 = 5 , r = = 2 , an
= a1
⋅ r −
5
1
52 n −
= ⋅
a n
1 1
n
33. a 1 = − 3 , r = =− , an
= a1
⋅ r −
−3 3
n−1 n−2
⎛ 1⎞ ⎛ 1⎞
a n =−3⎜− ⎟ = ⎜−
⎟
⎝ 3⎠ ⎝ 3⎠
1
n
34. a 1 = 4 , r = , an
= a1
⋅ r −
4
n−1 n−2
⎛1⎞ ⎛1⎞
a n = 4⎜ ⎟ = ⎜ ⎟
⎝ 4 ⎠ ⎝ 4 ⎠
35.
36.
n 1
an
= a1
⋅ r −
243 = a ⋅ −3
1
1
( )
5
( )
243 = a1
−3
243 =−243a
− 1 = a
Therefore,
a = a ⋅ r −
1
a n
n 1
n 1
2−1
⎛1
⎞
7 = a1
⎜ ⎟
⎝3
⎠
1
7 = a1
3
21 = a
1
Therefore,
a n
6−1
=−− ( 3) n
−1
n
⎛1
⎞
= 21⎜ ⎟
⎝ 3 ⎠
−1
.
1
1
1
28.
29.
30.
a =− 1, r =− 2, n=
10
a
1
10
1
8
−
( ) ( )
10 1 9
=−1⋅ − 2 = −1⋅ − 2 =−1( − 512) = 512
a = 0.4, r = 0.1, n=
8
a
1
7
−
( ) ( )
8 1 7
= 0.4⋅ 0.1 = 0.4 0.1 = 0.00000004
a = 0.1, r = 10, n=
7
a
7−1
( )
= 0.1⋅ 10 = 0.1 10 = 100,000
14
n
31. a 1 = 7 , r = = 2 , an
= a1
⋅ r −
7
1
7 2 n −
= ⋅
a n
6
1
37.
2 1
4−1 3
a4 a1⋅
r r
= = = r
2−1
a a ⋅ r r
2 1575
r = = 225
7
r = 225 = 15
a = a ⋅r
n
1
1
1
n−1
7 = a ⋅15
7 = 15a
7
a1
=
15
Therefore,
2−1
a n
2
7 15
n−1 7 15
n−2
= ⋅ = ⋅ .
15
1255
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