12-4
12-4Geometric
GeometricSequences
Sequencesand
andSeries
Series
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
12-4 Geometric Sequences and Series
Warm Up
Simplify.
1.
2.
Evaluate.
3. (–2)8
256
Solve for x.
5.
Holt Algebra 2
4.
96
12-4 Geometric Sequences and Series
Objectives
Find terms of a geometric sequence,
including geometric means.
Find the sums of geometric series.
Holt Algebra 2
12-4 Geometric Sequences and Series
Vocabulary
geometric sequence
geometric mean
geometric series
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
Holt Algebra 2
12-4 Geometric Sequences and Series
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 .
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
Holt Algebra 2
12-4 Geometric Sequences and Series
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,
Differences
Ratios
93,
–7
93
100
86,
–7
79
–7
86 79
93 86
It could be arithmetic, with d = –7.
Holt Algebra 2
12-4 Geometric Sequences and Series
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,
Differences
Ratios
It is neither.
Holt Algebra 2
90,
–90
60,
–30
1
2
15
–45
1
3
1
4
12-4 Geometric Sequences and Series
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
Ratios
1
1
1
5
5
5
It could be geometric, with
Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 1a
Determine whether the sequence could be
geometric or arithmetic. If possible, find the
common ratio or difference.
Differences
Ratios
It could be geometric with
Holt Algebra 2
12-4 Geometric Sequences and Series
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
Differences
1.3
–0.4
0.9
0.5
–0.4 –0.4
Ratio
It could be arithmetic, with r = –0.4.
Holt Algebra 2
12-4 Geometric Sequences and Series
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, . . .
Differences
Ratios
It is neither.
Holt Algebra 2
18
14
10
12-4 Geometric Sequences and Series
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.
nth term
an = an–1r
First term
Holt Algebra 2
Common
ratio
12-4 Geometric Sequences and Series
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.
Holt Algebra 2
12-4 Geometric Sequences and Series
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
a2
12
r=
=
=4
a1
3
Holt Algebra 2
12-4 Geometric Sequences and Series
Example 2 Continued
Step 2 Write a rule, and evaluate for n = 7.
an = a1 r
n–1
a7 = 3(4)7–1
= 3(4096) = 12,288
The 7th term is 12,288.
Holt Algebra 2
General rule
Substitute 3 for a1,7 for n,
and 4 for r.
12-4 Geometric Sequences and Series
Check Extend the sequence.
a4 = 192
Given
a5 = 192(4) = 768
a6 = 768(4) = 3072
a7 = 3072(4) = 12,288 
Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2a
Find the 9th term of the geometric sequence.
Step 1 Find the common ratio.
Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2a Continued
Step 2 Write a rule, and evaluate for n = 9.
an = a1 r
n–1
General rule
Substitute
n, and
The 9th term is
Holt Algebra 2
.
for a1, 9 for
for r.
12-4 Geometric Sequences and Series
Check It Out! Example 2a Continued
Check Extend the sequence.
Given
a6 =
a7 =
a8 =
a9 =
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2b Continued
Step 2 Write a rule, and evaluate for n = 9.
an = a1 r
n–1
a9 = 0.001(10)9–1
General rule
Substitute 0.001 for a1,
9 for n, and 10 for r.
= 0.001(100,000,000) = 100,000
The 7th term is 100,000.
Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2b Continued
Check Extend the sequence.
a5 = 10
Given
a6 = 10(10) = 100
a7 = 100(10) = 1,000
a8 = 1,000(10) = 10,000
a9 = 10,000(10) = 100,000
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
324 = 36r2
9 = r2
3 = r
Holt Algebra 2
Substitute 324 for a5 and 36 for a3.
Divide both sides by 36.
Take the square root of both sides.
12-4 Geometric Sequences and Series
Example 3 Continued
Step 2 Find a1.
Consider both the positive and negative values for r.
an = a1r
n-1
36 = a1(3)3 - 1
4 = a1
Holt Algebra 2
an = a1r
n-1
or 36 = a1(–3)3 - 1
4 = a1
General rule
Use a3 = 36
and r = 3.
12-4 Geometric Sequences and Series
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
an = 4(3)n - 1 or
an = 4(–3)n - 1
Substitute a1 and r.
a8 = 4(3)8 - 1
a8 = 4(–3)8 - 1
Evaluate for n = 8.
a8 = 8748
a8 = –8748
The 8th term is 8748 or –8747.
Holt Algebra 2
12-4 Geometric Sequences and Series
Caution!
When given two terms of a sequence, be
sure to consider positive and negative
values for r when necessary.
Holt Algebra 2
12-4 Geometric Sequences and Series
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
5=r
Holt Algebra 2
Substitute –40 for a5 and –8 for a4.
Divide both sides by –8.
12-4 Geometric Sequences and Series
Check It Out! Example 3a Continued
Step 2 Find a1.
an = a1r
n-1
–8 = a1(5)4 - 1
–0.064 = a1
Holt Algebra 2
General rule
Use a5 = –8 and r = 5.
12-4 Geometric Sequences and Series
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.
Holt Algebra 2
12-4 Geometric Sequences and Series
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
±0.25 = r
Holt Algebra 2
Divide both sides by 768.
Take the square root.
12-4 Geometric Sequences and Series
Check It Out! Example 3b Continued
Step 2 Find a1.
Consider both the positive and negative values for r.
an = a1r
768 = a1
(0.25)2 - 1
3072 = a1
Holt Algebra 2
n-1
an = a1r
or 768 = a1
n-1
General rule
Use a2= 768 and
r = 0.25.
(–0.25)2 - 1
–3072 = a1
12-4 Geometric Sequences and Series
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
3072(–0.25)n - 1
Substitute for
a1 and r.
a7 = 3072(0.25)7 - 1
a7 = 3072(–0.25)7 - 1
a7 = 0.75
a7 = 0.75
Evaluate for
n = 7.
an =
3072(0.25)n - 1
Holt Algebra 2
or an =
12-4 Geometric Sequences and Series
Check It Out! Example 3b Continued
an = a1r
an =
n-1
–3072(0.25)n - 1
a7 = –3072(0.25)7 - 1
a7 = –0.75
an = a1r
or an =
n-1
a7 = –3072(–0.25)7 - 1 Evaluate for
n = 7.
a7 = –0.75
The 7th term is 0.75 or –0.75.
Holt Algebra 2
Substitute for
a1 and r.
–3072(–0.25)n - 1
12-4 Geometric Sequences and Series
Geometric means are the terms between any two
nonconsecutive terms of a geometric sequence.
Holt Algebra 2
12-4 Geometric Sequences and Series
Example 4: Finding Geometric Means
Find the geometric mean of
and
Use the formula.
Holt Algebra 2
.
12-4 Geometric Sequences and Series
Check It Out! Example 4
Find the geometric mean of 16 and 25.
Use the formula.
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
Holt Algebra 2
12-4 Geometric Sequences and Series
Holt Algebra 2
12-4 Geometric Sequences and Series
Example 5A: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
S8 for 1 + 2 + 4 + 8 + 16 + ...
Step 1 Find the common ratio.
Holt Algebra 2
12-4 Geometric Sequences and Series
Example 5A Continued
Step 2 Find S8 with a1 = 1, r = 2, and n = 8.
Sum
formula
Substitute.
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Check Use a graphing
calculator.
12-4 Geometric Sequences and Series
Example 5B: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
Step 1 Find the first term.
Holt Algebra 2
12-4 Geometric Sequences and Series
Example 5B Continued
Step 2 Find S6.
Sum
formula
Substitute.
= 1(1.96875) ≈ 1.97
Holt Algebra 2
Check Use a graphing
calculator.
12-4 Geometric Sequences and Series
Check It Out! Example 5a
Find the indicated sum for each geometric series.
S6 for
Step 1 Find the common ratio.
Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 5a Continued
Step 2 Find S6 with a1 = 2, r =
, and n = 6.
Sum formula
Substitute.
Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 5b
Find the indicated sum for each geometric series.
Step 1 Find the first term.
Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 5b Continued
Step 2 Find S6.
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
4=n–1
Equate the exponents.
5=n
Solve for n.
Holt Algebra 2
12-4 Geometric Sequences and Series
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.
Holt Algebra 2
12-4 Geometric Sequences and Series
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
Holt Algebra 2
12-4 Geometric Sequences and Series
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, ….
–128
3. Find the 9th term of the geometric sequence
with a2 = 0.3 and a6 = 0.00003.
0.00000003
Holt Algebra 2
12-4 Geometric Sequences and Series
Lesson Quiz: Part II
4. Find the geometric mean of
and 18. 3
5. Find the indicated sum for the geometric series
40
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?
40
Holt Algebra 2