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How can a geometric sequence be described?ESSENTIAL QUESTION

L E S S O N

10.3 Geometric Sequences

Writing General Rules for Geometric SequencesIn a geometric sequence, the ratio of consecutive terms is constant. The constant ratio is called the common ratio, often represented by r.

Makers of Japanese swords in the 1400s repeatedly folded and hammered the metal to form layers. The folding process increased the strength of the sword.

The table shows how the number of layers depends on the number of folds. Write a recursive rule and an explicit rule for the geometric sequence represented by the table.

Number of Folds n 1 2 3 4 5

Number of Layers f(n) 2 4 8 16 32

STEP 1 Find the common ratio by calculating the ratios of consecutive terms.

4 _ 2 = 2 8 _ 4 = 2

16 __ 8 = 2 32 __ 16 = 2

The common ratio r is 2.

STEP 2 Write a recursive rule for the sequence.

The first term is 2, so f(1) = 2.

All terms after the first term are the product of the previous term and the common ratio: f(2) = f(1) · 2, f(3) = f(2) · 2, f(4) = f(3) · 2, …

The recursive rule is stated by providing the first term and the rule for successive terms.

f(1) = 2

f(n) = f(n - 1) · 2 for n ≥ 2

Makers of Japanese swords in the 1400s repeatedly folded and hammered the

EXAMPLE 1

This can be read as “each term in the sequence after the first term is equal to the previous term times two.”

COMMON CORE F.LE.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Also, A.SSE.3c, F.BF.1, F.BF.1a, F.BF.2, F.IF.3, F.IF.8b, F.LE.1a

COMMON CORE F.BF.2

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Lesson 10.3 353

STEP 3 Write an explicit rule for the sequence by writing each term as the

product of the first term and a power of the common ratio.

Generalize the results from the table: f(n) = 2 · 2 n - 1 .

REFLECT

1. Draw Conclusions How can you use properties of exponents to simplify

the explicit rule found in Example 1?

2. Justify Reasoning Explain why the sequence 4, 12, 36, 108, 324, …

appears to be a geometric sequence.

3. What If? A geometric sequence has a common ratio of 5. The 6th term of

the sequence is 30. What is the 7th term? What is the 5th term? Explain.

4. Communicate Mathematical Ideas The first term of a geometric

sequence is 81 and the common ratio is 1 _ 3 . Explain how you could find the

4th term of the sequence.

5. What is the recursive rule for the sequence f(n) = 5 (4) n - 1 ?

n f(n)

1 2(2) 0 = 2

2 2(2) 1 = 4

3 2 (2) 2 = 8

4 2 (2) 3 = 16

5 2 (2) 4 = 32

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354 Unit 2B

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General Rules for Geometric SequencesUse the geometric sequence 6, 24, 96, 384, 1536, … to help you write a recursive

rule and an explicit rule for any geometric sequence. For the general rules, the

values of n are consecutive integers starting with 1.

A Find the common ratio.

Numbers Algebra

6, 24, 96, 384, 1536, … f(1), f(2), f(3), f(4), f(5), …

Common ratio = 4 Common ratio = r

B Write a recursive rule.

Numbers Algebra

f(1) = 6 and Given f(1),

f(n) = f(n - 1) · 4 for n ≥ 2 f(n) = f(n - 1) · r for n ≥ 2

C Write an explicit rule.

Numbers Algebra

f(n) = 6 · 4 n - 1 f(n) = f(1) · r n - 1

EXPLORE ACTIVITY

6. Write a recursive rule and an explicit rule for the geometric sequence

represented by the table.

7. Write a recursive rule and an explicit rule for the geometric sequence

128, 32, 8, 2, 0.5, … .

YOUR TURN

n 1 2 3 4 5

f(n) 2 6 18 54 162

COMMON CORE F.BF.2

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Lesson 10.3 355

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Writing a Geometric Sequence Given Two TermsThe explicit and recursive rules for a geometric sequence can also be written in

subscript notation. In subscript notation, the subscript indicates the position of

the term in the sequence. a 1 , a 2 , and a 3 are the first, second, and third terms of a

sequence, respectively. In general, a n is the nth term of a sequence.

The shutter speed settings on a camera form a geometric sequence where a n is the shutter speed in seconds and n is the setting number. The fifth setting

on the camera is 1 __ 60

second, and the seventh setting on the camera is 1 __ 15

second.

Write an explicit rule for the sequence using subscript notation.

STEP 1 Identify the given terms in the sequence.

a 5 = 1 __ 60

The fifth term of the sequence is 1 ___

60 .

a 7 = 1 __ 15

The seventh term of the sequence is 1 ___

15 .

STEP 2 Find the common ratio.

a 7 = a 6 · r Write the recursive rule for a 7 .

a 6 = a 5 · r Write the recursive rule for a 6 .

a 7 = a 5 · r · r Substitute the expression for a 6 into the rule for a 7 .

1 __ 15

= 1 __ 60

· r 2 Substitute 1 ___

15 for a 7 and 1

___ 60

for a 5 .

4 = r 2 Multiply both sides by 60.

2 = r Definition of positive square root

STEP 3 Find the first term of the sequence.

a n = a 1 · r n - 1 Write the general explicit rule.

1 __ 60

= a 1 · 2 5 - 1 Substitute 1 ___

60 for a n , 2 for r, and 5 for n.

1 __ 60

= a 1 · 16 Simplify.

1 ___

960 = a 1 Divide both sides by 16.

STEP 4 Write the explicit rule.

a n = a 1 · r n - 1 Write the general explicit rule.

a n = 1 ___

960 · 2 n - 1 Substitute 1

____ 960

for a 1 and 2 for r.

Therefore, a n = 1 ___

960 · 2 n - 1 .

EXAMPLE 2

When finding the common ratio, why can you ignore the negative square root of 4 when

solving 4 = r 2 ?

COMMON CORE F.LE.2

Math TalkMathematical Practices

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Relating Geometric Sequences and Exponential FunctionsA geometric sequence is equivalent to an exponential function with a domain

that is restricted to the positive integers. For an exponential function of the form

f(n) = ab n , recall that a represents the initial value and b is the common ratio.

Compare this to f(n) = f(1) · r n - 1 , where f(1) represents the initial value and r is

the common ratio.

The graph shows the heights to which a ball

bounces after it is dropped. Write an explicit rule

for the sequence of bounce heights.

STEP 1 Represent the sequence in a table.

STEP 2 Examine the sequence to determine whether it is geometric. The

sequence is geometric because each term is the product of 0.8 and

the previous term. The common ratio is 0.8.

STEP 3 Write an explicit rule for the sequence.

f(n) = f(1) · r n -1 Write the general rule.

f(n) = 100 · 0.8 n - 1 Substitute 100 for f (1) and 0.8 for r.

The sequence has the rule f(n) = 100 · 0.8 n - 1 , where n is the

bounce number and f(n) is the bounce height.

EXAMPLE 3

n 1 2 3 4

f(n) 100 80 64 51.2

8. The third term of a geometric sequence is 1 __ 54

. The fifth term of the

sequence is 1 _ 6 . All terms of the sequence are positive numbers. Write an

explicit rule for the sequence using subscript notation.

YOUR TURN

9. The number of customers f(n) projected to

come into a new store in month number n

is represented by the following table.

Write an explicit rule for the sequence.

YOUR TURN

(2, 80)

f(n)

n

(3, 64)(4, 51.2)

2 4

40

80

120B

ou

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he

igh

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m)

Bounce number

Ball Bounces

6O

(1,100)

n 1 2 3 4

f(n) 1000 1500 2250 3375

COMMON CORE F.LE.2

Lesson 10.3 357

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Guided Practice

1. The table shows the beginning-of-month balances, rounded to the nearest

cent, in Marla’s saving account for the first few months after she made an

initial deposit in the account. (Example 1)

Month n 1 2 3 4

Account balance ($) f(n) 2000 2010.00 2020.05 2030.15

a. Explain how you know that the sequence of account balances is a

geometric sequence.

b. Write recursive and explicit rules for the sequences of account balances.

Recursive rule: f(1) = , f(n) = ‧

for n ≥ 2

Explicit rule: f(n) = ‧

2. Write a recursive rule and an explicit rule for the geometric sequence 9, 27,

81, 243. (Example 1)

27 ___ 9

= 81 ___ 27

= 243 ___ 81

=

Recursive rule:

Explicit rule:

3. Write an explicit rule for the geometric sequence with terms a 2 = 12 and

a 4 = 192. Assume that the common ratio r is positive. (Example 2)

4. Write an explicit rule for the geometric sequence with terms a 3 = 1600 and

a 5 = 256. Assume that the common ratio is positive. (Example 2)

5. How can you write the explicit rule for a geometric sequence if you know the

recursive rule for the sequence?

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358 Unit 2B

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