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This ppt. gives an introduction to Geometric Sequence and Series

1.
9-1

9-1 Geometric

GeometricSequences

Sequences

Warm Up

Lesson Presentation

Lesson Quiz

Holt

Holt

Holt McDougal

Algebra 1Algebra

McDougal Algebra11

9-1 Geometric

GeometricSequences

Sequences

Warm Up

Lesson Presentation

Lesson Quiz

Holt

Holt

Holt McDougal

Algebra 1Algebra

McDougal Algebra11

2.
9-1 Geometric Sequences

Warm Up

Find the value of each expression.

1. 25 32 2. 2–5

3. –34 –81 4. (–3)4 81

5. (0.2)3 0.008 6. 7(–4)2 112

7. 8. 12(–0.4)3 –0.768

Holt McDougal Algebra 1

Warm Up

Find the value of each expression.

1. 25 32 2. 2–5

3. –34 –81 4. (–3)4 81

5. (0.2)3 0.008 6. 7(–4)2 112

7. 8. 12(–0.4)3 –0.768

Holt McDougal Algebra 1

3.
9-1 Geometric Sequences

Objectives

Recognize and extend geometric

sequences.

Find the nth term of a geometric

sequence.

Holt McDougal Algebra 1

Objectives

Recognize and extend geometric

sequences.

Find the nth term of a geometric

sequence.

Holt McDougal Algebra 1

4.
9-1 Geometric Sequences

Vocabulary

geometric sequence

common ratio

Holt McDougal Algebra 1

Vocabulary

geometric sequence

common ratio

Holt McDougal Algebra 1

5.
9-1 Geometric Sequences

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.

Holt McDougal Algebra 1

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.

Holt McDougal Algebra 1

6.
9-1 Geometric Sequences

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.

1 2 3 4 Position

3 6 12 24 Term

a1 a2 a3 a4

To find a term in a geometric sequence, multiply

the previous term by r.

Holt McDougal Algebra 1

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.

1 2 3 4 Position

3 6 12 24 Term

a1 a2 a3 a4

To find a term in a geometric sequence, multiply

the previous term by r.

Holt McDougal Algebra 1

7.
9-1 Geometric Sequences

Holt McDougal Algebra 1

Holt McDougal Algebra 1

8.
9-1 Geometric Sequences

Writing Math

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.

Holt McDougal Algebra 1

Writing Math

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.

Holt McDougal Algebra 1

9.
9-1 Geometric Sequences

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.

Holt McDougal Algebra 1

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.

Holt McDougal Algebra 1

10.
9-1 Geometric Sequences

Example 1A Continued

Find the next three terms in the geometric

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.

Holt McDougal Algebra 1

Example 1A Continued

Find the next three terms in the geometric

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.

Holt McDougal Algebra 1

11.
9-1 Geometric Sequences

Example 1B: Extending Geometric Sequences

Find the next three terms in the geometric

sequence.

Step 1 Find the value of r by dividing each term

by the one before it.

The value

– of r is .

Holt McDougal Algebra 1

Example 1B: Extending Geometric Sequences

Find the next three terms in the geometric

sequence.

Step 1 Find the value of r by dividing each term

by the one before it.

The value

– of r is .

Holt McDougal Algebra 1

12.
9-1 Geometric Sequences

Helpful Hint

When the terms in a geometric sequence

alternate between positive and negative, the

value of r is negative.

Holt McDougal Algebra 1

Helpful Hint

When the terms in a geometric sequence

alternate between positive and negative, the

value of r is negative.

Holt McDougal Algebra 1

13.
9-1 Geometric Sequences

Example 1B Continued

Find the next three terms in the geometric

sequence.

Step 2 Multiply each term by to find the next

three terms.

The next three terms are

Holt McDougal Algebra 1

Example 1B Continued

Find the next three terms in the geometric

sequence.

Step 2 Multiply each term by to find the next

three terms.

The next three terms are

Holt McDougal Algebra 1

14.
9-1 Geometric Sequences

Check It Out! Example 1a

Find the next three terms in the geometric

sequence.

5, –10, 20,–40,…

Step 1 Find the value of r by dividing each term

by the one before it.

5 –10 20 –40

The value of

r is –2.

Holt McDougal Algebra 1

Check It Out! Example 1a

Find the next three terms in the geometric

sequence.

5, –10, 20,–40,…

Step 1 Find the value of r by dividing each term

by the one before it.

5 –10 20 –40

The value of

r is –2.

Holt McDougal Algebra 1

15.
9-1 Geometric Sequences

Check It Out! Example 1a Continued

Find the next three terms in the geometric

sequence.

5, –10, 20,–40,…

Step 2 Multiply each term by –2 to find the next

three terms.

–40 80 –160 320

(–2) (–2) (–2)

The next three terms are 80, –160, and 320.

Holt McDougal Algebra 1

Check It Out! Example 1a Continued

Find the next three terms in the geometric

sequence.

5, –10, 20,–40,…

Step 2 Multiply each term by –2 to find the next

three terms.

–40 80 –160 320

(–2) (–2) (–2)

The next three terms are 80, –160, and 320.

Holt McDougal Algebra 1

16.
9-1 Geometric Sequences

Check It Out! Example 1b

Find the next three terms in the geometric

sequence.

512, 384, 288,…

Step 1 Find the value of r by dividing each term

by the one before it.

512 384 288

The value of

r is 0.75.

Holt McDougal Algebra 1

Check It Out! Example 1b

Find the next three terms in the geometric

sequence.

512, 384, 288,…

Step 1 Find the value of r by dividing each term

by the one before it.

512 384 288

The value of

r is 0.75.

Holt McDougal Algebra 1

17.
9-1 Geometric Sequences

Check It Out! Example 1b Continued

Find the next three terms in the geometric

sequence.

512, 384, 288,…

Step 2 Multiply each term by 0.75 to find the next

three terms.

288 216 162 121.5

0.75 0.75 0.75

The next three terms are 216, 162, and 121.5.

Holt McDougal Algebra 1

Check It Out! Example 1b Continued

Find the next three terms in the geometric

sequence.

512, 384, 288,…

Step 2 Multiply each term by 0.75 to find the next

three terms.

288 216 162 121.5

0.75 0.75 0.75

The next three terms are 216, 162, and 121.5.

Holt McDougal Algebra 1

18.
9-1 Geometric Sequences

To find the output an of a geometric sequence

when n is a large number, you need an equation,

or function rule.

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.

Holt McDougal Algebra 1

To find the output an of a geometric sequence

when n is a large number, you need an equation,

or function rule.

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.

Holt McDougal Algebra 1

19.
9-1 Geometric Sequences

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

Holt McDougal Algebra 1

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

Holt McDougal Algebra 1

20.
9-1 Geometric Sequences

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.

Holt McDougal Algebra 1

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.

Holt McDougal Algebra 1

21.
9-1 Geometric Sequences

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.

Holt McDougal Algebra 1

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.

Holt McDougal Algebra 1

22.
9-1 Geometric Sequences

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

The value of r is

–3.

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.

Holt McDougal Algebra 1

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

The value of r is

–3.

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.

Holt McDougal Algebra 1

23.
9-1 Geometric Sequences

Caution

When writing a function rule for a sequence with

a negative common ratio, remember to enclose r

in parentheses. –212 ≠ (–2)12

Holt McDougal Algebra 1

Caution

When writing a function rule for a sequence with

a negative common ratio, remember to enclose r

in parentheses. –212 ≠ (–2)12

Holt McDougal Algebra 1

24.
9-1 Geometric Sequences

Check It Out! Example 2

What is the 8th term of the sequence 1000,

500, 250, 125, …?

1000 500 250 125

The value of r is .

an = a1rn–1 Write the formula.

a8 = 1000( )8–1 Substitute 1000 for a1,8 for n, and

for r.

Simplify the exponent.

= 7.8125 Use a calculator.

The 8th term of the sequence is 7.8125.

Holt McDougal Algebra 1

Check It Out! Example 2

What is the 8th term of the sequence 1000,

500, 250, 125, …?

1000 500 250 125

The value of r is .

an = a1rn–1 Write the formula.

a8 = 1000( )8–1 Substitute 1000 for a1,8 for n, and

for r.

Simplify the exponent.

= 7.8125 Use a calculator.

The 8th term of the sequence is 7.8125.

Holt McDougal Algebra 1

25.
9-1 Geometric Sequences

Example 3: Application

A ball is dropped from a

tower. The table shows Bounce Height (cm)

the heights of the balls

1 300

bounces, which form a

geometric sequence. 2 150

What is the height of the 3 75

6th bounce?

300 150 75

The value of r is

0.5.

Holt McDougal Algebra 1

Example 3: Application

A ball is dropped from a

tower. The table shows Bounce Height (cm)

the heights of the balls

1 300

bounces, which form a

geometric sequence. 2 150

What is the height of the 3 75

6th bounce?

300 150 75

The value of r is

0.5.

Holt McDougal Algebra 1

26.
9-1 Geometric Sequences

Example 3 Continued

an = a1rn–1 Write the formula.

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.

= 9.375 Use a calculator.

The height of the 6th bounce is 9.375 cm.

Holt McDougal Algebra 1

Example 3 Continued

an = a1rn–1 Write the formula.

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.

= 9.375 Use a calculator.

The height of the 6th bounce is 9.375 cm.

Holt McDougal Algebra 1

27.
9-1 Geometric Sequences

Check It Out! Example 3

The table shows a car’s

value for 3 years after it

Year Value ($)

is purchased. The values

form a geometric 1 10,000

sequence. How much 2 8,000

will the car be worth in

the 10th year? 3 6,400

10,000 8,000 6,400

The value of r is

0.8.

Holt McDougal Algebra 1

Check It Out! Example 3

The table shows a car’s

value for 3 years after it

Year Value ($)

is purchased. The values

form a geometric 1 10,000

sequence. How much 2 8,000

will the car be worth in

the 10th year? 3 6,400

10,000 8,000 6,400

The value of r is

0.8.

Holt McDougal Algebra 1

28.
9-1 Geometric Sequences

Check It Out! Example 3

an = a1rn–1 Write the formula.

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.

Holt McDougal Algebra 1

Check It Out! Example 3

an = a1rn–1 Write the formula.

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.

Holt McDougal Algebra 1

29.
9-1 Geometric Sequences

Lesson Quiz: Part I

Find the next three terms in each geometric

sequence.

1. 3, 15, 75, 375,… 1875; 9375; 46,875

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? 13.9968

4. What is the 15th term of the sequence 4, –8, 16,

–32, 64…? 65,536

Holt McDougal Algebra 1

Lesson Quiz: Part I

Find the next three terms in each geometric

sequence.

1. 3, 15, 75, 375,… 1875; 9375; 46,875

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? 13.9968

4. What is the 15th term of the sequence 4, –8, 16,

–32, 64…? 65,536

Holt McDougal Algebra 1

30.
9-1 Geometric Sequences

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

Holt McDougal Algebra 1

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

Holt McDougal Algebra 1