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9.3 Geometric Sequences

Essential Question: How can you write

a rule for a geometric sequence?

Key Vocabulary

Geometric sequence

The ratio of any term to the previous term is a

constant (multiplied).

Common Ratio

The constant ratio of a geometric sequence.

EXAMPLE 1 Identify geometric sequences

Tell whether the sequence is geometric.

a. 4, 10, 18, 28, 40, . . . b. 625, 125, 25, 5, 1, . . .

Geometric Sequences

a1 : the first term

r : common ration

1

1 )( n

n raa

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EXAMPLE 2 Write a rule for the nth term

Write a rule for the nth term of the sequence. Then find a7.

a. 4, 20, 100, 500, . . .

b. 152, –76, 38, –19, . . .

Summary of Steps to Find the Explicit Rule

for Geometric Sequences

Find r by dividing the given terms and rootingby the spaces between terms

If you are given r you may skip this step

Find a1 by plugging in an, n, and r into the equation an = a1 ∙ (r)n-1 Remember an is the answer given, n is the little subscript number If you are given a1 you may skip this step

Fill in a1 and r into the equation an = a1 ∙ (r)n-1

EXAMPLE 3 Write a rule given a term and common ratio

One term of a geometric sequence is a4 =12. The

common ratio is r = 2.

Write a rule for the nth term.

EXAMPLE 4 Write a rule given two terms

Two terms of a geometric sequence are a3 = –48 and

a6 = 3072. Find a rule for the nth term.

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GUIDED PRACTICE for Examples 2, 3 and 4

Write a rule for the nth term of the geometric sequence.

Then find a8.

2. 3, 15, 75, 375, . . .

3. a6 = –96, r = 2

4. a2 = –12, a4 = – 3

Remember…

When finding r, you must keep your pieces in

order

If the sequence is getting larger than |r| >1

Ex) 2, -3, 5, -4, etc.

If the sequence is getting smaller than |r| < 1

Ex) ½, -1/4, 1/3,

1/5, etc.

Divide the bigger “n” term by the smaller “n”

term even if it results in a fraction

Ex) a3 = 48, a6 = 6 means 6÷48 = 1/8 over 3 terms

so cube root 1/8 to get r = ½

MORE PRACTICE

Write a rule for the nth term of the geometric sequence.

1. 128, 64, 32, 16, . . .

2. a4 = 54, r = 3

MORE PRACTICE

Write a rule for the nth term of the geometric sequence.

3. a2 = 50, a4 = 2

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Switching between Explicit and Recursive Form

The easiest way to switch between forms is to

create a table

Ex) f(n) = 2(3)n-1 is in ___________ form

To switch to ____________ form make a table

n f(n)

1

2

3

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Switching between Recursive and Explicit Form

The easiest way to switch between forms is to

create a table

Ex) f(1) = 5; f(n) = f(n-1) ∙ 2; n ≥ 2 is in ___________

form

To switch to ____________ form make a table

n f(n)

1

2

3

4

Comparing explicit and recursive

forms of geometric sequences Look at the last two slides and see if you can

find a relationship between the forms.

Write explicit form from a1 = 2, an = an-1 ∙ 4