8.3
Geometric Sequences
and Series
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What You Should Learn
•
Recognize, write, and find the nth terms of
geometric sequences.
•
Find nth partial sums of geometric sequences.
•
Find sums of infinite geometric series.
•
Use geometric sequences to model and solve
real-life problems.
2
Geometric Sequences
3
Geometric Sequences
We know that a sequence whose consecutive terms have a
common difference is an arithmetic sequence.
In this section, you will study another important type of
sequence called a geometric sequence. Consecutive
terms of a geometric sequence have a common ratio.
4
Geometric Sequences
5
Example 1 – Examples of Geometric Sequences
a. The sequence whose nth term is 2n is geometric. For this
sequence, the common ratio between consecutive terms
is 2.
2, 4, 8, 16, . . . , 2n, . . .
Begin with n = 1.
b. The sequence whose nth term is 4(3n) is geometric. For
this sequence, the common ratio between consecutive
terms is 3.
12, 36, 108, 324, . . . , 4(3n), . . .
Begin with n = 1.
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Example 1 – Examples of Geometric Sequences
cont’d
c. The sequence whose nth term is
is geometric. For
this sequence, the common ratio between consecutive
terms is
Begin with n = 1.
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Geometric Sequences
8
The Sum of a Finite
Geometric Sequence
9
The Sum of a Finite Geometric Sequence
The formula for the sum of a finite geometric sequence is
as follows.
10
Example 6 – Finding the Sum of a Finite Geometric Sequence
Find the following sum.
Solution:
See next slide
11
Example 6 – Solution
cont’d
Because
a1 = 4(0.3),
r = 0.3,
and
n = 12
you can apply the formula for the sum of a finite geometric
sequence to obtain
Formula for sum of a finite
geometric sequence
Substitute 4(0.3) for a1, 0.3
for r, and 12 for n.
 1.71.
Use a calculator.
12
Geometric Series
Read slides #14 & 15, but only copy the fomula on slide 15.
13
Geometric Series
The sum of the terms of an infinite geometric sequence is
called an infinite geometric series or simply a geometric
series.
The formula for the sum of a finite geometric sequence
can, depending on the value of r, be extended to produce a
formula for the sum of an infinite geometric series.
Specifically, if the common ratio has the property that
|r| < 1
then it can be shown that r n becomes arbitrarily close to
zero as n increases without bound.
14
Geometric Series
Consequently,
This result is summarized as follows.
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Example 7 – Finding the Sum of an Infinite Geometric Series
Find each sum.
a.
b. 3 + 0.3 + 0.03 + 0.003 + . . .
Solution:
a.
= 10
16
Example 7 – Solution
cont’d
b. 3 + 0.3 + 0.03 + 0.003 + . . .
= 3 + 3(0.1) + 3(0.1)2 + 3(0.2)3 + . . .
 3.33
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