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Holt McDougal Algebra 1
9-1 Geometric Sequences
Geometric
Sequence

Review
Find the ratio of the second number to
the first number.
1. 2 and 8 2. 12 and 4
3. –3 and 9 4. -49 and 7
4
5. 1 and
–3
2
7. -5 and -10 a
2

Warm Up
Find the value of each expression.
1. 25 2. 2–5
3. –34 4. (–3)4
32
5. (0.2)3 6. 7(–4)2
–81 81
112
0.008
7. 8. 12(–0.4)3
–0.768

Recognize and extend geometric
sequences.
Find the nth term of a geometric
sequence.
Objectives

geometric sequence
common ratio
Vocabulary

The table shows the heights of a bungee
jumper’s bounces.
The height of the bounces shown in the table
above form a geometric sequence. In a
geometric sequence, the ratio of successive
terms is the same number r, called the
common ratio.

Geometric sequences can be thought of as
functions. The term number, or position in the
sequence, is the input, and the term itself is the
output.
a1 a2 a3 a4
1 2 3 4 Position
3 6 12 24 Term
To find a term in a geometric sequence, multiply
the previous term by r.

The variable a is often used to represent terms in
a sequence. The variable a4 (read “a sub 4”)is the
fourth term in a sequence.
Writing Math

Example 1A: Extending Geometric Sequences
Find the next three terms in the geometric
sequence.
1, 4, 16, 64,…
Step 1 Find the value of r by dividing each term
by the one before it.
1 4 16 64
The value of r is 4.

Example 1A Continued
sequence.
1, 4, 16, 64,…
Step 2 Multiply each term by 4 to find the next
three terms.
64 256 1024 4096
4 4 4
The next three terms are 256, 1024, and 4096.

Example 1B: Extending Geometric Sequences
sequence.
The value
of r is .
–

When the terms in a geometric sequence
alternate between positive and negative, the
value of r is negative.
Helpful Hint

Example 1B Continued
sequence.
Step 2 Multiply each term by to find the next
three terms.
The next three terms are

Check It Out! Example 1a
sequence.
5, –10, 20,–40,…
5 –10 20 –40
The value of
r is –2.

sequence.
Step 2 Multiply each term by –2 to find the next
three terms.
(–2) (–2) (–2)
The next three terms are 80, –160, and 320.
Check It Out! Example 1a Continued
5, –10, 20,–40,…
–40 80 –160 320

Check It Out! Example 1b
sequence.
512, 384, 288,…
512 384 288
The value of
r is 0.75.

Check It Out! Example 1b Continued
sequence.
512, 384, 288,…
Step 2 Multiply each term by 0.75 to find the next
three terms.
0.75 0.75 0.75
The next three terms are 216, 162, and 121.5.
288 216 162 121.5

The pattern in
the table shows
that to get the
nth term,
multiply the first
term by the
common ratio
raised to the
power n – 1.
To find the output an of a geometric sequence
when n is a large number, you need an equation,
or function rule.

If the first term of a geometric sequence is a1,
the nth term is an , and the common ratio is r,
then
an = a1rn–1
nth term 1st term Common ratio

Example 2A: Finding the nth Term of a Geometric
Sequence
The first term of a geometric sequence is 500,
and the common ratio is 0.2. What is the 7th
term of the sequence?
an = a1rn–1
Write the formula.
a7 = 500(0.2)7–1
Substitute 500 for a1,7 for n, and
0.2 for r.
= 500(0.2)6 Simplify the exponent.
= 0.032 Use a calculator.
The 7th term of the sequence is 0.032.

Example 2B: Finding the nth Term of a Geometric
Sequence
For a geometric sequence, a1 = 5, and r = 2.
Find the 6th term of the sequence.
an = a1rn–1
Write the formula.
a6 = 5(2)6–1
Substitute 5 for a1,6 for n, and 2
for r.
= 5(2)5 Simplify the exponent.
= 160
The 6th term of the sequence is 160.

Example 2C: Finding the nth Term of a Geometric
Sequence
What is the 9th term of the geometric
sequence 2, –6, 18, –54, …?
2 –6 18 –54
an = a1rn–1 Write the formula.
a9 = 2(–3)9–1 Substitute 2 for a1,9 for n, and –3
for r.
= 2(–3)8 Simplify the exponent.
= 13,122 Use a calculator.
The 9th term of the sequence is 13,122.
The value of r is
–3.

When writing a function rule for a sequence with
a negative common ratio, remember to enclose r
in parentheses. –212 ≠ (–2)12
Caution

Check It Out! Example 2
What is the 8th term of the sequence 1000,
500, 250, 125, …?
1000 500 250 125
Simplify the exponent.
The 8th term of the sequence is 7.8125.
The value of r is .
Substitute 1000 for a1,8 for n, and
for r.
a8 = 1000( )8–1

Example 3: Application
A ball is dropped from a
tower. The table shows
the heights of the balls
bounces, which form a
geometric sequence.
What is the height of the
6th bounce?
Bounce Height (cm)
1 300
2 150
3 75
300 150 75
The value of r is
0.5.

Example 3 Continued
a6 = 300(0.5)6–1 Substitute 300 for a1, 6 for n, and
0.5 for r.
= 300(0.5)5 Simplify the exponent.
The height of the 6th bounce is 9.375 cm.

The table shows a car’s
value for 3 years after it
is purchased. The values
form a geometric
sequence. How much
will the car be worth in
the 10th year?
Year Value ($)
1 10,000
2 8,000
3 6,400
10,000 8,000 6,400
The value of r is
0.8.

a6 = 10,000(0.8)10–1 Substitute 10,000 for a1
,10 for n,
and 0.8 for r.
= 10,000(0.8)9 Simplify the exponent.
= 1,342.18 Use a calculator.
In the 10th year, the car will be worth $1342.18.

Lesson Quiz: Part I
Find the next three terms in each geometric
sequence.
1. 3, 15, 75, 375,…
2.
3. The first term of a geometric sequence is 300
and the common ratio is 0.6. What is the 7th
term of the sequence?
4. What is the 15th term of the sequence 4, –8, 16,
–32, 64…?
1875; 9375; 46,875
65,536
13.9968

Lesson Quiz: Part II
Find the next three terms in each geometric
sequence.
5. The table shows a car’s value for three years
after it is purchased. The values form a
geometric sequence. How much will the car be
worth after 8 years?
$5570.39 Year Value ($)
1 18,000
2 15,300
3 13,005

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9-1 Geometric Sequences.ppt

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