Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12: Sequences; Induction; the Binomial Theorem
38.
39.
3 1
6−1 5
a6 a1
⋅ r r
= = = r
3−1 2
a a ⋅ r r
r
1
3 81
1
3
1 1
= = ⋅ 3 =
81 27
1 1
r = 3 =
27 3
n−1
a = a ⋅r
n 1
3−1
1 ⎛1⎞
= a1
⋅ ⎜ ⎟
3 ⎝3⎠
1 1
= a1
3 9
3 = a
1
Therefore,
a n
3
n−1 n−2
⎛1⎞ ⎛1⎞
= 3⎜ ⎟ = ⎜ ⎟
⎝ 3 ⎠ ⎝ 3 ⎠
1
a1
= , r = 2
4
⎛
n
n
1−r
⎞ 1⎛1−2 ⎞ 1
Sn
= a1
⎜ 1 2
1 r ⎟
= 4⎜ 1 2 ⎟
= − −
⎝ − ⎠ ⎝ − ⎠ 4
1
( 2
n
= − 1 )
4
.
n
( )
42.
a = 4, r = 3
S
1
n
⎛
n n n
1−r
⎞ ⎛1−3 ⎞ ⎛1−3
⎞
= a1
⎜ = 4 = 4
1−r
⎟ ⎜ 1−3 ⎟ ⎜ −2
⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
n n
( ) ( )
=−21− 3 = 23 −1
43. a1
= − 1, r = 2
S
n
⎛
n
n
1−r
⎞ ⎛1−2
⎞
= a1
⎜ =− 1 = 1−2
1−r
⎟ ⎜ 1−2
⎟
⎝ ⎠ ⎝ ⎠
3
44. a1
= 2, r =
5
S
n
⎡
n
n
⎛3⎞ ⎤ ⎡ ⎛3⎞
⎤
n ⎢1−
⎥ ⎢1−
⎥
⎛1− r ⎞ ⎜ ⎟ ⎜ ⎟
5 5
= a1
2
⎝ ⎠
2
⎝ ⎠
⎢ ⎥ ⎢ ⎥
⎜ 1 r ⎟
= =
⎢ 3 ⎥ ⎢ 2 ⎥
⎝ − ⎠ 1
⎛ ⎞
⎢ − 5
⎥ ⎢ ⎜ ⎟ ⎥
⎢⎣ ⎥⎦ ⎢⎣ ⎝ 5 ⎠ ⎥⎦
⎡
n
⎛3
⎞ ⎤
= 51 ⎢ −⎜
⎟ ⎥
⎢⎣
⎝5
⎠ ⎥⎦
45. Using the sum of the sequence feature:
n
40.
3 1
a1
= = , r = 3
9 3
⎛
n n n
1−r
⎞ 1⎛1−3 ⎞ 1⎛1−3
⎞
Sn
= a1
⎜ 1 r ⎟
= 3⎜ 1 3 ⎟
=
3⎜ 2 ⎟
⎝ − ⎠ ⎝ − ⎠ ⎝ − ⎠
1 n 1 n
=− ( 1− 3 ) = ( 3 −1)
6 6
46. Using the sum of the sequence feature:
2 2
41. a1
= , r =
3 3
S
n
⎡
n
⎛2
⎞ ⎤
n ⎢1−
⎥
⎛1− r ⎞ 2
⎜ ⎟
3
= a
⎝ ⎠
1
⎢ ⎥
⎜ 1 r ⎟
=
3⎢ 2 ⎥
⎝ − ⎠
⎢ 1−
3
⎥
⎢⎣
⎥⎦
⎡
n
⎛2
⎞ ⎤
⎢1−
2
⎜ ⎟ ⎥
n
3 ⎡ ⎛2⎞
⎤
= ⎢ ⎝ ⎠ ⎥ = 21 ⎢ − ⎥
3⎢ 1 ⎥ ⎜ ⎟
⎢ ⎝3⎠
⎥
⎢
⎣ ⎦
3
⎥
⎢⎣
⎥⎦
47. Using the sum of the sequence feature:
48. Using the sum of the sequence feature:
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