Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12 Test
2. a1 an
an
− 1
= 4; = 3 + 2
a2 = 3a1+ 2= 3( 4)
+ 2=
14
a3 = 3a2
+ 2= 3( 14)
+ 2=
44
a4 = 3a3
+ 2= 3( 44)
+ 2=
134
a = 3a
+ 2 = 3 134 + 2 = 404
5 4
( )
The first five terms of the sequence are 4, 14, 44,
134, and 404.
3
3. ∑ ( 1)
4.
5.
k = 1
k + 1 ⎛k
+ 1⎞
− ⎜ 2 ⎟
⎝ k ⎠
11 + 1+ 1 21 + 2+ 1 31 + 3+
1
= ( − 1
⎛ ⎞
) ( 1
⎛ ⎞
) ( 1
⎛ ⎞
⎜ )
2
⎟+ − ⎜
2
⎟+ − ⎜ 2 ⎟
⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ 3 ⎠
2⎛2⎞ 3⎛3⎞ 4⎛4⎞
= ( − 1) ⎜ ( 1) ( 1)
1
⎟+ − ⎜
4
⎟+ − ⎜
9
⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
3 4 61
= 2 − + = 4 9 36
4
∑
k
⎡⎛2
⎞ ⎤
⎢⎜
− k ⎥
3
⎟
⎢⎣⎝
⎠ ⎥⎦
k = 1
⎡ 1 ⎤ ⎡ 2 ⎤ ⎡ 3 ⎤ ⎡ 4
⎤
2 2 2 2
( ) ( ) ( ) ( )
= ⎢ − 1 2 3 4
3 ⎥ + ⎢ −
3 ⎥ + ⎢ −
3 ⎥ + ⎢ −
3 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎛2 ⎞ ⎛4 ⎞ ⎛ 8 ⎞ ⎛16
⎞
= ⎜ − 1 2 3 4
3
⎟+ ⎜ −
9
⎟+ ⎜ −
27
⎟+ ⎜ −
81
⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
1 14 73 308
=− − − −
3 9 27 81
680
=−
81
2 3 4 11
− + − + ... +
5 6 7 14
Notice that the signs of each term alternate, with
the first term being negative. This implies that
the general term will include a power of − 1 .
Also note that the numerator is always 1 more
than the term number and the denominator is 4
more than the term number. Thus, each term is in
k ⎛ k + 1 ⎞
the form ( −1)
⎜
k + 4
⎟. The last numerator is 11
⎝ ⎠
which indicates that there are 10 terms.
10
2 3 4 11 k ⎛ k + 1⎞
− + − + ... + = ( 1)
5 6 7 14
∑ − ⎜
k + 4
⎟
⎝ ⎠
k = 1
6. 6,12,36,144,...
12 − 6 = 6 and 36 − 12 = 24
The difference between consecutive terms is not
constant. Therefore, the sequence is not
arithmetic.
7.
12 = 2 and 36
6 12
= 3
The ratio of consecutive terms is not constant.
Therefore, the sequence is not geometric.
1 4
n
a n = − ⋅
2
1 n 1 n−1
a − ⋅4 − ⋅4 ⋅4
n 2 2
= = = 4
a 1 n−1 1 n−1
− ⋅4 − ⋅4
n−1 2 2
Since the ratio of consecutive terms is constant,
the sequence is geometric with common ratio
1 1
r = 4 and first term a 1 =− ⋅ 4 =− 2 .
2
The sum of the first n terms of the sequence is
given by
n
1−
r
Sn
= a1
⋅
1−
r
n
1−
4
=−2⋅
1 − 4
2
( 1 4
n
= − )
3
8. −2, −10, −18, − 26,...
−10 −( − 2)
= − 8 , ( )
−18 − − 10 = − 8 ,
−26 −( − 18)
= − 8
The difference between consecutive terms is
constant. Therefore, the sequence is arithmetic
with common difference d =− 8 and first term
a 1 = − 2 .
an
= a1 + ( n−1)
d
= − 2+ ( n −1)( −8)
=−2− 8n
+ 8
= 6−8n
The sum of the first n terms of the sequence is
given by
n
Sn
= ( a+
an)
2
n
= ( − 2+ 6−8n)
2
n
= ( 4−8n)
2
= n 2−4n
( )
1281
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