Geometric Sequences and Series
Section 9-3
2
Objectives
• Recognize, write, and find nth terms of geometric sequences
• Find the nth partial sums of geometric sequences
• Find the sum of an infinite geometric sequence
3
Definition of a Geometric Sequence
• A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.
4
An infinite sequence is a function whose domain is the set of positive integers.
a1, a2, a3, a4, . . . , an, . . .
The first three terms of the sequence an = 2n2 are
a1 = 2(1)2 = 2
a2 = 2(2)2 = 8
a3 = 2(3)2 = 18.
finite sequence
terms
5
A sequence is geometric if the ratios of consecutive terms are the same.
2, 8, 32, 128, 512, . . .
geometric sequence
The common ratio, r, is 4.
82
4
328
4
12832
4
512128
4
6
General Term of a Geometric Sequence
• The nth term (the general term) of a geometric sequence with the first term a1 and common ratio r is
• an = a1 r n-1
7
The nth term of a geometric sequence has the form
an = a1rn - 1
where r is the common ratio of consecutive terms of the sequence.
15, 75, 375, 1875, . . . a1 = 15
The nth term is: an = 15(5)n-1.
75 515
r
a2 = 15(5)
a3 = 15(52)
a4 = 15(53)
8
Example: Find the 9th term of the geometric sequence
7, 21, 63, . . .
a1 = 7
The 9th term is 45,927.
21 37
r
an = a1rn – 1 = 7(3)n – 1
a9 = 7(3)9 – 1 = 7(3)8
= 7(6561) = 45,927
9
The Sum of the First n Terms of a Geometric Sequence
r
raS
n
n
1
)1(1
The sum, Sn, of the first n terms of a geometric sequence is given by
in which a1 is the first term and r is the common ratio.
10
Example
5314404
)5314411(4
)3(1
))3(1(4
1
)1(
1
)1(
12121
12
1
r
raS
r
raS
n
n
• Find the sum of the first 12 terms of the geometric sequence: 4, -12, 36, -108, ...Solution:
11
The sum of the first n terms of a sequence is represented by summation notation.
1 2 3 41
n
i ni
a a a a a a
index of summation
upper limit of summation
lower limit of summation
5
1
4n
n
1 2 3 4 54 4 4 4 4 4 16 64 256 1024 1364
12
The sum of a finite geometric sequence is given by
11 1
1
1 .1
n nin
i
rS a r ar
5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?
n = 8
a1 = 5
1
81 11
221
5n
nrS ar
5210r
1 25651 2 2555
1 1275
13
The Sum of an Infinite Geometric Series
If -1<r<1, then the sum of the infinite geometric series
a1+a1r+a1r2+a1r3+…
in which a1 is the first term and r s the common ration is given by
r
aS
11
If |r|>1, the infinite series does not have a sum.
14
Example: Find the sum of
1
1a
Sr
1 13 13 9
13
r
3
1 13
3 31 413 3
The sum of the series is 9 .4
3 934 4
15
...16
1
8
1
4
1
2
1
r
aS
11
21
21
1S
121
21
S
16
...64
3
32
3
16
3
8
3
r
aS
11
21
83
1 S
4
1
23
83
S
17
Homework
• WS 13-5