Pre-Algebra

Pre-Algebra

12-2 Geometric Sequences

Learn to find terms in a geometric sequence.

Pre-Algebra


In a geometric sequence, the ratio of one term to the next is always the same. This ratio is called the common ratio. The common ratio is multiplied by each term to get the next term.

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Determine if the sequence could be geometric. If so, give the common ratio.

A. 1, 5, 25, 125, 625, …

Example 1A: Identifying Geometric Sequences

The sequence could be a geometric with a common ratio of 5.

Divide each term by the term before it.

1 5 25 125 625, . . .

5555

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B. 1, 3, 9, 12, 15, …

Example 1B: Identifying Geometric Sequences

The sequence is not geometric.


1 3 9 12 15, . . .

54

4333

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C. 81, 27, 9, 3, 1, . . .

Example 1C: Identifying Geometric Sequences

The sequence could be geometric with a common ratio of .1

3


81 27 9 3 1, . . .

13

1313

13

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D. –3, 6, –12, 24, –48

Example 1D: Identifying Geometric Sequences

The sequence could be geometric with a common ratio of –2.


–3 6 –12 24 –48, . . .

–2–2–2–2

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FINDING THE nth TERM OF A GEOMETRIC SEQUENCE

The nth term an of a geometric sequence with common ratio r is

an = a1rn–1.

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Find the given term in the geometric sequence.

A. 11th term: –2, 4, –8, 16, . . .

Example 2A: Finding a Given Term of a Geometric Sequence

an = a1rn–1

a11 = –2(–2)10 = –2(1024) = –2048

r = = –2 4–2

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B. 9th term: 100, 70, 49, 34.3, . . .

Example 2B: Finding a Given Term of a Geometric Sequence

an = a1rn–1

a9 = 100(0.7)8 = 100(0.05764801) = 5.764801

r = = 0.7 70 100

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C. 10th term: 0.01, 0.1, 1, 10, . . .

Example 2C: Finding a Given Term of a Geometric Sequence

an = a1rn–1

a10 = 0.01(10)9 = 0.01(1,000,000,000) = 10,000,000

r = = 10 0.10.01

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D. 7th term: 1000, 200, 40, 8, . . .

Example 2D: Finding a Given Term of a Geometric Sequence

an = a1rn–1

r = = 2001000

15

a7 = 1000( )6 = 1000( )= , or 0.064

15

8 125

1 15,625

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A. 2, 10, 50, 250, 1250, . . .

Try This: Example 1A



2 10 50 250 1250, . . .

5555

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B. 1, 1, 1, 1, 1, . . .

Try This: Example 1B



1 1 1 1 1, . . .

1111

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an = a1rn–1.

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A. 12th term: -2, 4, -8, 16, . . .

Try This: Example 2A

an = a1rn–1

a12 = –2(–2)11 = –2(–2048) = 4096

r = = –2 4–2

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B. 11th term: 100, 70, 49, 34.3, . . .

Try This: Example 2B

a11 = 100(0.7)10 = 100(0.0282475249) ≈ 2.825

an = a1rn–1

r = = 0.7 70100

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Siobhan sells computers. She has the option of earning (1) $50 per sale or (2) $1 for the first sale, $2 for the second sale, $4 for the third sale and so on, where each sale is worth twice as much as the previous sale. If Siobhan estimates that she can sell 10 computers a week, which option should she choose?

Example 3: Money Application

If Siobhan chooses $50 per sale, she will get a total of 10($50) = $500.

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Example 3 Continued

a10 = ($1)(2)9 = ($1)(512) = $512

Option 1 gives Siobhan more money in the beginning, but option 2 gives her a larger total amount.

If Siobhan chooses the second option, her earnings for just the 10th sale will be more that the total of all the earnings in option 1.

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an = a1rn–1.

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C. 2, 4, 12, 24, 96, . . .

Try This: Example 1C

The sequence is not geometric.


2 4 12 24 96, . . .

4232

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D. 1, 2, 4, 8, 16, . . .

Try This: Example 1D

The sequence could be geometric with a common ratio of 2.

1 2 4 8 16, . . .

2222


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C. 5th term: 0.01, 0.1, 1, 10, . . .

Try This: Example 2C

a5 = 0.01(10)4 = 0.01(10,000) = 100

an = a1rn–1

r = = 10 0.10.01

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D. 12th term: 1000, 200, 40, 8, …

Try This: Example 2D

an = a1rn–1

r = = 2001000

15

a5 = 1000 ( )4 = 1000( )= , or 1.6

15

85

1 625

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A gumball machine at the mall has 932 gumballs. If 19 gumballs are bought each day, how many gumballs will be left in the machine on the 7th day?

Try This: Example 3

a7 = (932)(0.98)6 ≈ (932)(0.89) ≈ 829

an = a1rn–1

r = ≈ 0.98913932

n = 7 a1 = 932

Pre-Algebra

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