Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12 Review Exercises
50.
8
12! = 479,001,600 = 4.790016×
10
18
20! ≈ 2.432902008×
10
25
25! ≈ 1.551121004×
10
⎛12 ⎞ ⎛ 1 ⎞
12! ≈ 2⋅12π ⎜ ⎟ ⎜1+
⎟
⎝ e ⎠ ⎝ 12 ⋅ 12 − 1 ⎠
≈ 479,013,972.4
⎛20 ⎞ ⎛ 1 ⎞
20! ≈ 2⋅20π ⎜ ⎟ ⎜1+
⎟
⎝ e ⎠ ⎝ 12 ⋅ 20 − 1 ⎠
18
≈ 2.43292403×
10
⎛25 ⎞ ⎛ 1 ⎞
25! ≈ 2⋅25π ⎜ ⎟ ⎜1+
⎟
⎝ e ⎠ ⎝ 12 ⋅ 25 − 1 ⎠
25
≈ 1.551129917×
10
Chapter 12 Review Exercises
1.
2.
3.
4.
12
20
25
11+ 3 4 2 2+
3 5
a1 = ( − 1) =− , a2
= ( − 1) = ,
1+ 2 3 2+
2 4
3 3+ 3 6 4 4+
3 7
a3 = ( − 1) =− , a4
= ( − 1) = ,
3+ 2 5 4+
2 6
5 5+
3 8
a5
= ( − 1) =− 5 + 2 7
11 +
b1
= ( −1) (2 ⋅ 1+ 3) = 5,
2+
1
b2
= ( −1) (2⋅ 2+ 3) =−7,
3+
1
b3
= ( −1) (2⋅ 3+ 3) = 9,
4+
1
b4
= ( −1) (2⋅ 4+ 3) =−11,
5+
1
b = ( −1) (2⋅ 5+ 3) = 13
5
1 2 3
1 2 2 2 3 2
2 2 2 4 2 8
c = = = 2, c = = = 1, c = = ,
1 1 2 4 3 9
4 5
2 16 2 32
c4 = = = 1, c
2 5 = =
2
4 16 5 25
1 2 3 4 5
1 = e = , 2 = e , 3 = e , e 4 = ,
e
5 =
d e d d d d
1 2 3 4 5
2 2 4
a = 3, a = ⋅ 3 = 2, a = ⋅ 2 = ,
3 3 3
2 4 8 2 8 16
a4 = ⋅ = , a5
= ⋅ =
3 3 9 3 9 27
5. 1 2 3
1 1 1
= 4, =− ⋅ 4 =− 1, =− ⋅− 1 = ,
4 4 4
1 1 1 1 1 1
a4 =− ⋅ =− , a5
=− ⋅− =
4 4 16 4 16 64
6. a1 a2 a3
a = 2, a = 2 − 2 = 0, a = 2 − 0 = 2,
7. 1 2 3
a
= 2− 2= 0, a = 2− 0=
2
4 5
8. a1 a2 a3
9.
a
= − 3, = 4 + ( − 3) = 1, = 4 + 1 = 5,
= 4+ 5= 9, a = 4+ 9=
13
4 5
4
∑
k = 1
(4k
+ 2)
( 4 1 2) ( 4 2 2) ( 4 3 2) ( 4 4 2)
( 6) ( 10) ( 14) ( 18)
= ⋅ + + ⋅ + + ⋅ + + ⋅ +
= + + +
= 48
3
2 2 2 2
10. ∑ (3 − k ) = ( 3 − 1 ) + ( 3 − 2 ) + ( 3 −3
)
k = 1
( 2) ( 1) ( 6)
= + − + −
=−5
1 1 1 1 k + 1 ⎛1⎞
− + − +⋅⋅⋅+ = − ⎜ ⎟
2 3 4 13
⎝k
⎠
11. 1 ∑ ( 1)
13
k = 1
2 3 4 n+ 1 n 1
2 2 2 2 ⎛
k+
2 ⎞
12. 2 + + + +⋅⋅⋅+ =
2 3
n ∑
k
3 3 3 3 ⎜
k = 0 3 ⎟
⎝ ⎠
n+
1⎛
k
2 ⎞
= ∑
⎜ k −1
k = 1 3 ⎟
⎝ ⎠
13. { a } { n 5}
n
= + Arithmetic
d = ( n+ 1+ 5) − ( n+ 5) = n+ 6−n− 5=
1
n
n
Sn
= [ 6+ n+ 5 ] = ( n+
11)
2 2
14. { b } { 4n
3}
n
= + Arithmetic
d = (4( n+ 1) + 3) − (4n+
3)
= 4n+ 4+ 3−4n− 3=
4
n
Sn
= + +
2
n
= (4n
+ 10)
2
= n +
[ 7 4n
3]
( 2n
5)
1273
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