Geometric Sequences and Series

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Geometric Sequences and Series
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
Holt McDougal Algebra 2
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Warm Up Simplify. 1. 2. 3. (2)8 4.
Solve for x. 5.
96
Evaluate.
256
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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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Serena Williams was the winner out of 128 players
who began the 2003 Wimbledon Ladies Singles
Championship. After each match, the winner
continues to the next round and the loser is
eliminated from the tournament. This means that
after each round only half of the players remain.
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The number of players remaining after each round
can be modeled by a geometric sequence. In a
geometric sequence, the ratio of successive terms
is a constant called the common ratio r (r ? 1)
. For the players remaining, r is .
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Recall that exponential functions have a
common ratio. When you graph the ordered pairs
(n, an) of a geometric sequence, the points lie
on an exponential curve as shown. Thus, you can
think of a geometric sequence as an exponential
function with sequential natural numbers as the
domain.
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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, ...
100, 93, 86, 79
Differences 7 7 7
It could be arithmetic, with d 7.
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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, ...
180, 90, 60, 15
Differences 90 30 45
It is neither.
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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, ...
5, 1, 0.2, 0.04
Differences 4 0.8 0.16
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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, . . .
1.7 1.3 0.9 0.5
It could be arithmetic, with r 0.4.
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Check It Out! Example 1c
Determine whether each sequence could be
geometric or arithmetic. If possible, find the
common ratio or difference.
50, 32, 18, 8, . . .
50, 32, 18, 8, . . .
It is neither.
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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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You can also use an explicit rule to find the nth
term of a geometric sequence. Each term is the
product of the first term and a power of the
common ratio as shown in the table.
This pattern can be generalized into a rule for
all geometric sequences.
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Example 2 Finding the nth Term Given a Geometric
Sequence
Find the 7th term of the geometric sequence 3,
12, 48, 192, ....
Step 1 Find the common ratio.
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Example 2 Continued
Step 2 Write a rule, and evaluate for n 7.
an a1 r n1
General rule
Substitute 3 for a1,7 for n, and 4 for r.
a7 3(4)71
3(4096) 12,288
The 7th term is 12,288.
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Check Extend the sequence.
a4 192
Given
a5 192(4) 768
a6 768(4) 3072
a7 3072(4) 12,288
?
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Check It Out! Example 2a
Find the 9th term of the geometric sequence.
Step 1 Find the common ratio.
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Check It Out! Example 2a Continued
Step 2 Write a rule, and evaluate for n 9.
an a1 r n1
General rule
3
3
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Check It Out! Example 2a Continued
Check Extend the sequence.
Given
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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, . . .
Step 1 Find the common ratio.
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Check It Out! Example 2b Continued
Step 2 Write a rule, and evaluate for n 9.
an a1 r n1
General rule
Substitute 0.001 for a1, 9 for n, and 10 for r.
a9 0.001(10)91
0.001(100,000,000) 100,000
The 9th term is 100,000.
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Check It Out! Example 2b Continued
Check Extend the sequence.
Given
a5 10
a6 10(10) 100
a7 100(10) 1,000
a8 1,000(10) 10,000
a9 10,000(10) 100,000
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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.
a5 a3 r2
Simplify.
Substitute 324 for a5 and 36 for a3.
324 36r2
9 r2
Divide both sides by 36.
?3 r
Take the square root of both sides.
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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
General rule
36 a1(3)3 - 1
36 a1(3)3 - 1
Use a3 36 and r ?3.
or
4 a1
4 a1
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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
General rule
Substitute a1 and r.
an 4(3)n - 1
an 4(3)n - 1
or
a8 4(3)8 - 1
a8 4(3)8 - 1
Evaluate for n 8.
a8 8748
a8 8748
The 8th term is 8748 or 8748.
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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
Step 1 Find the common ratio.
a5 a4 r(5 4)
Use the given terms.
a5 a4 r
Simplify.
40 8r
Substitute 40 for a5 and 8 for a4.
5 r
Divide both sides by 8.
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Check It Out! Example 3a Continued
Step 2 Find a1.
an a1r n - 1
General rule
8 a1(5)4 - 1
Use a5 8 and r 5.
0.064 a1
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Check It Out! Example 3a Continued
Step 3 Write the rule and evaluate for a7.
an a1r n - 1
an 0.064(5)n - 1
Substitute for a1 and r.
a7 0.064(5)7 - 1
Evaluate for n 7.
a7 1,000
The 7th term is 1,000.
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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
Step 1 Find the common ratio.
a4 a2 r(4 2)
Use the given terms.
a4 a2 r2
Simplify.
48 768r2
Substitute 48 for a4 and 768 for a2.
0.0625 r2
Divide both sides by 768.
0.25 r
Take the square root.
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Check It Out! Example 3b Continued
Step 2 Find a1.
Consider both the positive and negative values
for r.
General rule
an a1r n - 1
an a1r n - 1
Use a2 768 and r ?0.25.
768 a1(0.25)2 - 1
768 a1(0.25)2 - 1
or
3072 a1
3072 a1
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Check It Out! Example 3b Continued
Step 3 Write the rule and evaluate for a7.
Consider both the positive and negative values
for r.
an a1r n - 1
an a1r n - 1
Substitute for a1 and r.
an 3072(0.25)n - 1
an 3072(0.25)n - 1
or
a7 3072(0.25)7 - 1
a7 3072(0.25)7 - 1
Evaluate for n 7.
a7 0.75
a7 0.75
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Check It Out! Example 3b Continued
an a1r n - 1
an a1r n - 1
Substitute for a1 and r.
an 3072(0.25)n - 1
an 3072(0.25)n - 1
or
a7 3072(0.25)7 - 1
a7 3072(0.25)7 - 1
Evaluate for n 7.
a7 0.75
a7 0.75
The 7th term is 0.75 or 0.75.
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Geometric means are the terms between any two
nonconsecutive terms of a geometric sequence.
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Example 4 Finding Geometric Means
Use the formula.
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Check It Out! Example 4
Find the geometric mean of 16 and 25.
Use the formula.
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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.
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 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.
1(1.96875) 1.97
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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.
Sum formula
Substitute.
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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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Example 6 Sports Application
An online video game tournament begins with 1024
players. Four players play in each game, and in
each game, only the winner advances to the next
round. How many games must be played to determine
the winner?
Step 1 Write a sequence.
Let n the number of rounds,
an the number of games played in the nth round,
and
Sn the total number of games played through n
rounds.
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Example 6 Continued
Step 2 Find the number of rounds required.
The final round will have 1 game, so substitute 1
for an.
Isolate the exponential expression by dividing by
256.
Equate the exponents.
4 n 1
Solve for n.
5 n
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Example 6 Continued
Step 3 Find the total number of games after 5
rounds.
Sum function for geometric series
341 games must be played to determine the winner.
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Check It Out! Example 6
A 6-year lease states that the annual rent for an
office space is 84,000 the first year and will
increase by 8 each additional year of the lease.
What will the total rent expense be for the
6-year lease?
? 616,218.04
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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.
geometric r 6
2. Find the 8th term of the geometric sequence
1, 2, 4, 8, .
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3. Find the 9th term of the geometric sequence
with a2 0.3 and a6 0.00003.
0.00000003
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Lesson Quiz Part II
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5. Find the indicated sum for the geometric series
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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?
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