9.3 Geometric Sequence

A sequence is geometric if the ratios of consecutive terms are the same.

That means if each term is found by multiplying the preceding term by the same number. This number is called the COMMON RATIO.

if

there is a number r (r ≠ 0) such that

r is called the COMMON RATIO.

6, 12, 24, 48, …

2, 10, 50, 250, 1250, …

800, 200, 50, , …

Examples of geometric sequences:

What is the common ratio?

6, 12, 24, 48, … r = 2 because

2, 10, 50, 250, 1250, …r=5 because

800, 200, 50, , …

r =

because =

What is the common ratio?

Is this a geometric sequence? If so, what is the common ratio?

36, 27, 18, … not a geometric sequence

Because and and

9, -6, 4, -, …Is a geometric sequence

and and

and so on.

Therefore, r =

Is this a geometric sequence? If so, what is the common ratio?

4, 12, 36, 108, …Is a geometric sequence

r = 3

4, 8, 16, 36, …Not a geometric sequence, but

Let’s look at the geometric sequence 6, 12, 24, 48, 96 , … in a different way

6, 12, 24, 48, 96,

,

6 , ( 6 ∙2 ) , ( 6∙2∙2 ) , ( 6∙2∙2∙2 ) , ( 6∙2∙2∙2∙2 )

6∙ 6∙ 6∙ 6∙ 6∙

From this example, we can derive a rule:

The nth Term of a Geometric Sequence

∙

where r is the common ratio and is the first term of the sequence

Find the first 4 terms of the geometric sequence.

−3 −3∙3= −9

−9∙3=−27

−27∙3=−81

Find the first 5 terms of the geometric sequence.

3

=3∙5=1515=15∙5=75

,

In this sequence, explain the position of these 2 terms: and

comes directly after

Find the first 5 terms of the geometric sequence.

,

= 5

Try This!

Find the first 5 terms of the sequence if and

Write an expression for the nth term of the geometric sequence.

Remember:

, n=4This means you are to find .

= −4∙= −4 125∙= −500

Write an expression for the nth term of the geometric sequence.Remember:

, n=9This means you are to find .

64=

64=

=

Try This!

Find of this sequence:2, 6, 18, …

2∙=2∙729=1458

Write an expression for for this geometric sequence:

4, 20, 100, 500, 2500, …

Remember:

What is 4

What is r?5

Substitute these values into

𝑎𝑛=4 ∙5𝑛−1

The book will simplify this, and here how it is done.

4∙=4∙∙

YOU TRY IT !Write an expression for for the geometric sequence5, −10, 20, −40, 80, …

Now you try the homework problems!!