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Section 9-3. Geometric Sequences and Series. 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. Definition of a Geometric Sequence. - PowerPoint PPT Presentation
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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