Pre-Algebra

12-2 Geometric Sequences12-2 Geometric Sequences

Pre-Algebra

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Pre-Algebra

12-2 Geometric Sequences

Warm Up1. Determine if the sequence could be arithmetic. If so, give the common difference. 100, 50, 25, 12.5, . . .

Find the given term in each arithmetic sequence.

2. 12th term: a1 = 30, d = 0.5

3. 55th term: 4, 28, 52, 76

no

35.5

1300

Pre-Algebra

12-2 Geometric Sequences

Pre-Algebra

12-2 Geometric Sequences

Problem of the Day

Two students begin counting by 3’s at the same time. One counts up from 0, and the other counts down from 120. If each says one number every second, will both students ever say the same number at the same time?yes (60)

Pre-Algebra

12-2 Geometric Sequences

Learn to find terms in a geometric sequence.

Pre-Algebra

12-2 Geometric Sequences

Vocabulary

geometric sequencecommon ratio

Pre-Algebra

12-2 Geometric Sequences

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.

Pre-Algebra

12-2 Geometric Sequences

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

A. 1, 5, 25, 125, 625, …

Additional 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

Pre-Algebra

12-2 Geometric Sequences

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

B. 1, 3, 9, 12, 15, …

Additional Example 1B: Identifying Geometric Sequences

The sequence is not geometric.

Divide each term by the term before it.

1 3 9 12 15, . . .

54

43

33

Pre-Algebra

12-2 Geometric Sequences

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

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

Additional Example 1C: Identifying Geometric Sequences

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

3

Divide each term by the term before it.

81 27 9 3 1, . . .

13

13

13

13

Pre-Algebra

12-2 Geometric Sequences

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

D. –3, 6, –12, 24, –48

Additional Example 1D: Identifying Geometric Sequences

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

Divide each term by the term before it.

–3 6 –12 24 –48, . . .

–2–2–2–2

Pre-Algebra

12-2 Geometric Sequences

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

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

Try This: Example 1A

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

Divide each term by the term before it.

2 10 50 250 1250, . . .

5555

Pre-Algebra

12-2 Geometric Sequences

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

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

Try This: Example 1B

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

Divide each term by the term before it.

1 1 1 1 1, . . .

1111

Pre-Algebra

12-2 Geometric Sequences

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

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

Try This: Example 1C

The sequence is not geometric.

Divide each term by the term before it.

2 4 12 24 96, . . .

4232

Pre-Algebra

12-2 Geometric Sequences

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

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

Divide each term by the term before it.

Pre-Algebra

12-2 Geometric Sequences

FINDING THE nth TERM OF A GEOMETRIC SEQUENCE

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

an = a1rn–1.

Pre-Algebra

12-2 Geometric Sequences

Find the given term in the geometric sequence.

A. 11th term: –2, 4, –8, 16, . . .

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

an = a1rn–1

a11 = –2(–2)10 = –2(1024) = –2048

r = = –2 4–2

Pre-Algebra

12-2 Geometric Sequences

Find the given term in the geometric sequence.

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

Additional 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

Pre-Algebra

12-2 Geometric Sequences

Find the given term in the geometric sequence.

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

Additional 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

Pre-Algebra

12-2 Geometric Sequences

Find the given term in the geometric sequence.

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

Additional 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

Pre-Algebra

12-2 Geometric Sequences

Find the given term in the geometric sequence.

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

Pre-Algebra

12-2 Geometric Sequences

Find the given term in the geometric sequence.

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

Pre-Algebra

12-2 Geometric Sequences

Find the given term in the geometric sequence.

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

Pre-Algebra

12-2 Geometric Sequences

Find the given term in the geometric sequence.

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

Pre-Algebra

12-2 Geometric Sequences

Tara 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 Tara estimates that she can sell 10 computers a week, which option should she choose?

Additional Example 3: Money Application

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

Pre-Algebra

12-2 Geometric Sequences

Additional Example 3 Continued

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

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

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

Pre-Algebra

12-2 Geometric Sequences

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

There will be about 829 gumballs in the machine after 7 days.

Pre-Algebra

12-2 Geometric Sequences

Lesson QuizDetermine if each sequence could be geometric. If so, give the common ratio.

1. 200, 100, 50, 25, 12.5, . . .

2. 4, 8, 12, 16, . . .

Find the given term in each geometric

sequence.

3. 7th term: , 1, 3, 9, . . .

4. 20th term: a1 = 800, r = 0.8

no

243

≈ 11.53

13

yes; 12