Find terms of a geometric sequence, including geometric means.

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Objectives
Find terms of a geometric sequence, including
geometric means. Find the sums of geometric
series.
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Vocabulary
geometric sequence geometric mean geometric series
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Example 1A Identifying Geometric Sequences
Determine whether the sequence could be geometric
or arithmetic. If possible, find the common ratio
or difference.
100, 93, 86, 79, ...
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Example 1B Identifying Geometric Sequences
Determine whether the sequence could be geometric
or arithmetic. If possible, find the common ratio
or difference.
180, 90, 60, 15, ...
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Example 1C Identifying Geometric Sequences
Determine whether the sequence could be geometric
or arithmetic. If possible, find the common ratio
or difference.
5, 1, 0.2, 0.04, ...
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Check It Out! Example 1a
Determine whether the sequence could be geometric
or arithmetic. If possible, find the common ratio
or difference.
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Check It Out! Example 1b
Determine whether the sequence could be geometric
or arithmetic. If possible, find the common ratio
or difference.
1.7, 1.3, 0.9, 0.5, . . .
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Each term in a geometric sequence is the product
of the previous term and the common ratio, giving
the recursive rule for a geometric sequence.
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Check It Out! Example 2b
Find the 9th term of the geometric sequence.
0.001, 0.01, 0.1, 1, 10, . . .
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Example 3 Finding the nth Term Given Two Terms
Find the 8th term of the geometric sequence with
a3 36 and a5 324.
Step 1 Find the common ratio.
a5 a3 r(5 3)
Use the given terms.
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Example 3 Continued
Step 2 Find a1.
Consider both the positive and negative values
for r.
an a1r n - 1
an a1r n - 1
36 a1(3)3 - 1
36 a1(3)3 - 1
or
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Example 3 Continued
Step 3 Write the rule and evaluate for a8.
Consider both the positive and negative values
for r.
an a1r n - 1
an a1r n - 1
an 4(3)n - 1
an 4(3)n - 1
or
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Caution!
When given two terms of a sequence, be sure to
consider positive and negative values for r when
necessary.
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Check It Out! Example 3a
Find the 7th term of the geometric sequence with
the given terms.
a4 8 and a5 40
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Check It Out! Example 3b
Find the 7th term of the geometric sequence with
the given terms.
a2 768 and a4 48
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The indicated sum of the terms of a geometric
sequence is called a geometric series. You can
derive a formula for the partial sum of a
geometric series by subtracting the product of Sn
and r from Sn as shown.
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Example 5A Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
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Example 5A Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
Step 1 Find the common ratio.
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Example 5A Continued
Step 2 Find S8 with a1 1, r 2, and n 8.
Check Use a graphing calculator.
Substitute.
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Example 5A Continued
Step 2 Find S8 with a1 1, r 2, and n 8.
Check Use a graphing calculator.
Substitute.
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Example 5B Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
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Example 5B Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
Step 1 Find the first term.
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Example 5B Continued
Step 2 Find S6.
Check Use a graphing calculator.
Substitute.
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Example 5B Continued
Step 2 Find S6.
Check Use a graphing calculator.
Substitute.
1(1.96875) 1.97
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Check It Out! Example 5a
Find the indicated sum for each geometric series.
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Check It Out! Example 5a
Find the indicated sum for each geometric series.
Step 1 Find the common ratio.
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Check It Out! Example 5a Continued
Step 2 Find S6 with a1 2, r , and n 6.
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Check It Out! Example 5a Continued
Step 2 Find S6 with a1 2, r , and n 6.
Sum formula
Substitute.
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Check It Out! Example 5b
Find the indicated sum for each geometric series.
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Check It Out! Example 5b
Find the indicated sum for each geometric series.
Step 1 Find the first term.
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Check It Out! Example 5b Continued
Step 2 Find S6.
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Check It Out! Example 5b Continued
Step 2 Find S6.
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Lesson Quiz Part I
1. Determine whether the sequence could be
geometric or arithmetic. If possible, find the
common ratio or difference.
2. Find the 8th term of the geometric sequence
1, 2, 4, 8, .
3. Find the 9th term of the geometric sequence
with a2 0.3 and a6 0.00003.
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Lesson Quiz Part II
5. Find the indicated sum for the geometric series
6. A math tournament begins with 81 students.
Students compete in groups of 3, with 1 person
from each trio going on to the next round until
there is 1 winner. How many matches must be
played in order to complete the tournament?