Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12: Sequences; Induction; the Binomial Theorem
93. 1, 1, 2, 3, 5, 8, 13
This is the Fibonacci sequence.
94. a. u1 = 1, u2 = 1, u3 = 2, u4 = 3, u5
= 5,
u6 = 8, u7 = 13, u8 = 21, u9 = 34, u10
= 55,
u = 89
b.
11
u2 1 u3
2 u4
3
= = 1, = = 2, = = 1.5,
u1 1 u2 1 u3
2
u5 5 u6
8
= ≈ 1.67, = = 1.6,
u4 3 u5
5
u7 13 u8
21
= = 1.625, = ≈1.615,
u6 8 u7
13
u9 34 u10
55
= ≈ 1.619, = ≈1.618,
u8 21 u9
34
u11
89
= ≈1.618
u 55
10
⎛
1+
5⎞
c. 1.618 ⎜The exact value is ⎟
⎝
2 ⎠
u1 1 u2
1 u3
2
d. = = 1, = = 0.5, = ≈ 0.667,
u2 1 u3 2 u4
3
u4
3 u5
5
= = 0.6, = = 0.625,
u5 5 u6
8
u6 8 u7
13
= ≈ 0.615, = ≈0.619,
u7 13 u8
21
u8 21 u9
34
= ≈ 0.618, = ≈0.618,
u9 34 u10
55
u10
55
= ≈0.618
u 89
11
1.3
c. f ( ) e
1.3 = ≈ 3.669296668
d. It will take n = 12 to approximate
f
1.3
( 1.3) e
96. a. ( 2.4)
= correct to 8 decimal places.
3
( 2.4)
∑
2.4
f e − −
− = ≈
b. ( 2.4)
k = 0
k!
( −2.4) ( −2.4) ( −2.4) ( −2.4)
+ + +
0 1 2 3
=
0! 1! 2! 3!
=−0.824
2.4 ( 2.4)
f e − −
− = ≈
2.4
c. f ( ) e −
6
∑
k = 0
k!
( −2.4) ( −2.4) ( −2.4)
= + + ... +
0 1 6
0! 1! 6!
= 0.1602688
− 2.4 = ≈ 0.0907179533
k
k
e. 0.618
⎛
2 ⎞
⎜The exact value is ⎟
⎝
1 + 5 ⎠
d. It will take n = 17 to approximate
f
2.4
( 2.4) e −
− = correct to 8 decimal places.
95. a. f ( 1.3)
b. f ( 1.3)
4
k
1.3 1.3
∑ k!
k = 0
= e ≈
0 1 4
1.3 1.3 1.3
= + + ... +
0! 1! 4!
≈ 3.630170833
7
k
1.3 1.3
∑ k!
k = 0
= e ≈
0 1 7
1.3 1.3 1.3
= + + ... +
0! 1! 7!
≈ 3.669060828
1244
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