Algebra1 Geometric Sequences

CONFIDENTIAL 1

Algebra1Algebra1

Geometric Geometric SequencesSequences

CONFIDENTIAL 2

Warm UpWarm Up

Write a function to describe each of the following graphs.

1) The graph of f (x) = x2 - 3 translated 7 units up

2) The graph of f (x) = 2x2 + 6 narrowed and translated 2 units down

1) f (x) = x2 + 4

2) f (x) = 3x2 + 4

CONFIDENTIAL 3

Bungee jumpers can use geometric sequences to calculate how high they will bounce.

The table shows the heights of a bungee jumper’s bounces.

Geometric SequencesGeometric Sequences

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.

CONFIDENTIAL 4

Find the next three terms in each geometric sequence.


A) 1, 3, 9, 27, …

Step1: Find the value of r by dividing each term by the one before it.

1 3 9 27

The value of r is 3.

3 = 3 9 = 3 27 = 31 3 9

CONFIDENTIAL 5

The next three terms are 81, 243, and 729.

Step2: Multiply each term by 3 to find the next three terms.

27 81 243 729

× 3 × 3 × 3

CONFIDENTIAL 6

B) -16, 4, -1, 1 , … 4


-16 4 -1 1 4

The value of r is -1. 4

4 = -1 -1 = -1 1/4 = -1-16 4 4 4 -1 4

CONFIDENTIAL 7

The next three terms are -1, 1, and 1. 16 64 256

Step2: Multiply each term by -1 to find the next 4 three terms.

1 -1 1 -14 16 64 256

× -1 4

× -1 4

× -1 4

CONFIDENTIAL 8

Now you try!


1a) 5, -10, 20, -40, …

1b) 512, 384, 288, …

1a) 80, -160, 3201b) 216, 162, 121.5

CONFIDENTIAL 9

Geometric sequences can be thought of as functions. The term number, or position in the sequence, is the input of the

function, and the term itself is the output of the function.

To find the output an of a geometric sequence when n is a large number, you need an equation, or function rule. Look for a pattern to find a function rule for the sequence above.

CONFIDENTIAL 10

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.

If the first term of a geometric sequence is a 1 , the nth term is a n , and the common ratio is r, then

nth term 1st term Common ratio

an = a1rn - 1

CONFIDENTIAL 11

Finding the nth Term of a Geometric SequenceFinding the nth Term of a Geometric Sequence

A) The first term of a geometric sequence is 128, and the common ratio is 0.5. What is the 10th term

of the sequence?

an = a1rn - 1 Write the formula.

Substitute 128 for a1 , 10 for n, and 0.5 for r.

Simplify the exponent.

Use a calculator.

a10 = (128)(0.5)10 - 1

a10 = (128)(0.5)9

a10 = 0.25

The 10th term of the sequence is 0.25.

CONFIDENTIAL 12

B) For a geometric sequence, a 1 = 8 and r = 3. Find the 5th term of this sequence.


Substitute 8 for a1 , 5 for n, and 3 for r.


Use a calculator.

a5 = (8)(3)5 - 1

a5 = (8)(3)4

a5 = 648

The 5th term of the sequence is 648.

CONFIDENTIAL 13

C) What is the 13th term of the geometric sequence 8, -16, 32, -64, … ?

8 -16 32 -64

The value of r is -2.

-16 = -2 32 = -2 -64 = -2 8 -16 32


CONFIDENTIAL 14


Substitute 8 for a1 , 13 for n, and -2 for r.


Use a calculator.

a13 = (8)(-2)13 - 1

a13 = (8)(-2)12

a13 = 32,768

The 13th term of the sequence is 32,768.

Step2: Plug the value of r in the following formula.

CONFIDENTIAL 15

Now you try!

2) What is the 8th term of the sequence 1000, 500, 250, 125, … ?

2) 7.8125

CONFIDENTIAL 16

Sports ApplicationSports Application

A bungee jumper jumps from a bridge. The diagram shows the bungee jumper’s height above the ground at the top of

each bounce. The heights form a geometric sequence. What is the bungee jumper’s height at the top of the 5th bounce?

The value of r is 0.4.

200 80 32

80 = 0.4 32 = 0.4200 80

CONFIDENTIAL 17




Use a calculator.

a5 = (200)(0.4)5 - 1

a5 = (200)(0.5)4

a5 = 5.12

The height of the 5th bounce is 5.12 feet.

CONFIDENTIAL 18

Now you try!

3) 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?

3) $1342.18

CONFIDENTIAL 19

Assessment

1) 2, 4, 8, 16, …


2) 400, 200, 100, 50, …

3) 4, -12, 36, -108, …

1) 32, 64, 1282) 25, 12.5, 6.253) 324, -972, 29164)-1250, 6250, -31,250

4) -2, 10, -50, 250, …

CONFIDENTIAL 20

5) The first term of a geometric sequence is 1, and the common ratio is 10. What is the 10th term of the sequence?

5) 1,000,000,0006) 3072

6) What is the 11th term of the geometric sequence 3, 6, 12, 24, … ?

CONFIDENTIAL 21

7) In the NCAA men’s basketball tournament, 64 teams compete in round 1. Fewer teams remain in each following

round, as shown in the graph, until all but one team have been eliminated. The numbers of teams in each round form a

geometric sequence. How many teams compete in round 5?

7) 4

CONFIDENTIAL 22

8) 20, 40,___,____ , …

8) 80, 1609) 2, , , 54

9) ___, 6, 18,___, …

Find the missing term(s) in each geometric sequence.

CONFIDENTIAL 23

Bungee jumpers can use geometric sequences to calculate how high they will bounce.

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.

Let’s review

CONFIDENTIAL 24



A) 1, 3, 9, 27, …


1 3 9 27

The value of r is 3.

3 = 3 9 = 3 27 = 31 3 9

CONFIDENTIAL 25

The next three terms are 81, 243, and 729.

Step2: Multiply each term by 3 to find the next three terms.

27 81 243 729

× 3 × 3 × 3

CONFIDENTIAL 26

Geometric sequences can be thought of as functions. The term number, or position in the sequence, is the input of the

function, and the term itself is the output of the function.

To find the output an of a geometric sequence when n is a large number, you need an equation, or function rule. Look for a pattern to find a function rule for the sequence above.

CONFIDENTIAL 27

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.

If the first term of a geometric sequence is a 1 , the nth term is a n , and the common ratio is r, then

nth term 1st term Common ratio

an = a1rn - 1

CONFIDENTIAL 28

Finding the nth Term of a Geometric SequenceFinding the nth Term of a Geometric Sequence

A) The first term of a geometric sequence is 128, and the common ratio is 0.5. What is the 10th term

of the sequence?




Use a calculator.

a10 = (128)(0.5)10 - 1

a10 = (128)(0.5)9

a10 = 0.25

The 10th term of the sequence is 0.25.

CONFIDENTIAL 29

Sports ApplicationSports Application

A bungee jumper jumps from a bridge. The diagram shows the bungee jumper’s height above the ground at the top of

each bounce. The heights form a geometric sequence. What is the bungee jumper’s height at the top of the 5th bounce?

The value of r is 0.4.

200 80 32

80 = 0.4 32 = 0.4200 80

CONFIDENTIAL 30




Use a calculator.

a5 = (200)(0.4)5 - 1

a5 = (200)(0.5)4

a5 = 5.12

The height of the 5th bounce is 5.12 feet.

CONFIDENTIAL 31

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Algebra1 Geometric Sequences