Geometric Sequences

Geometric Sequences

Another simple way of generating a sequence is to start with a number ¡°a¡± and repeatedly

multiply it by a fixed nonzero constant ¡°r¡±. This type of sequence is called a geometric

sequence.

Definition: A geometric sequence is a sequence of the form

a, ar , ar 2 , ar 3 , ar 4 , ...

The number a is the first term, and r is the common ratio of the sequence. The

nth term of a geometric sequence is given by

an = ar n ?1 .

The number r is called the common ratio because any two consecutive terms of the sequence

differ by a multiple of r, and it is found by dividing any term an +1 after the first by the preceding

term an . That is

r=

an +1

.

an

Is the Sequence Geometric?

Example 1: Determine whether the sequence is geometric. If it is geometric, find the common

ratio.

(a) 2, 8, 32, 128, ...

(b) 1, 2, 3, 5, 8, ...

Solution (a): In order for a sequence to be geometric, the ratio of any term to the

one that precedes it should be the same for all terms. If they are all

the same, then r, the common difference, is that value.

Step 1: First, calculate the ratios between each term and the one that precedes it.

8

=4

2

32

=4

8

128

=4

32

By: Crystal Hull

Example 1 (Continued):

Step 2: Now, compare the ratios. Since the ratio between each term and the one

that precedes it is 4 for all the terms, the sequence is geometric, and the

common ratio r = 4 .

Solution (b):

Step 1: Calculate the ratios between each term and the one that precedes it.

2

=1

1

3 3

=

2 2

5 5

=

3 3

8 8

=

5 5

Step 2: Compare the ratios. Since they are not all the same, the sequence is not

geometric.

Similar to an arithmetic sequence, a geometric sequence is determined completely by the first

term a, and the common ratio r. Thus, if we know the first two terms of a geometric sequence,

then we can find the equation for the nth term.

Finding the Terms of a Geometric Sequence:

Example 2: Find the nth term, the fifth term, and the 100th term, of the geometric sequence

1

determined by a = 6, r = .

3

Solution: To find a specific term of a geometric sequence, we use the formula

for finding the nth term.

Step 1: The nth term of a geometric sequence is given by

an = ar n ?1

So, to find the nth term, substitute the given values a = 6, r =

?1?

an = 6 ? ?

? 3?

1

into the formula.

3

n ?1

By: Crystal Hull

Example 2 (Continued):

Step 2: Now, to find the fifth term, substitute n = 5 into the equation for the nth

term.

5 ?1

?1?

a5 = 6 ? ?

?3?

?1?

= 6? 4 ?

?3 ?

6

=

81

2

=

27

Step 3: Finally, find the 100th term in the same way as the fifth term.

100 ?1

?1?

a5 = 6 ? ?

?3?

? 1 ?

= 6 ? 99 ?

?3 ?

2?3

= 99

3

2

= 98

3

Example 3: Find the common ratio, the fifth term and the nth term of the geometric sequence.

(a) ?1, 9, ? 81, 729, ...

1 t t2 t3

(b) , , , , ...

2 6 18 54

Solution (a): In order to find the nth term, we will first have to determine what a

and r are. We will then use the formula for finding the nth term of

a geometric sequence.

By: Crystal Hull

Example 3 (Continued):

Step 1: First, determine what a and r are. The number a is always the first term

of the sequence, so

a = ?1 .

The ratio between any term and the one that precedes it should be the same

because the sequence is geometric, so we can choose any pair to find the

common ratio r. If we choose the first two terms

9

?1

= ?9.

r=

Step 2: Since we are given the fourth term, we can multiply it by the common

ratio r = ?9 to get the fifth term.

a5 = a4 ? r

= 729 ( ?9 )

= ?6561

Step 3: Now, to find the nth term, substitute a = ?1, r = ?9 into the formula for

the nth term of a geometric sequence.

an = ar n ?1

= ( ?1)( ?9 )

= ? ( ?9 )

n ?1

n ?1

By: Crystal Hull

Example 3 (Continued):

Solution (b):

Step 1: Calculate a and r.

1

2

?t?

? ?

6

r=? ?

?1?

? ?

?2?

? t ?? 2 ?

= ? ?? ?

? 6 ?? 1 ?

t

=

3

a=

Step 2: The fifth term is the fourth term multiplied by the common ratio.

Therefore,

a5 = a4 ? r

? t3 ? ? t ?

= ? ?? ?

? 54 ? ? 3 ?

t4

=

162

1

t

Step 3: Now, substitute a = , r = into the formula for the nth term.

2

3

? 1 ?? t ?

an = ? ? ? ?

? 2 ?? 3 ?

n ?1

Partial Sums of a Geometric Sequence:

We can start developing a formula for the sum of the first n terms of a geometric sequence, Sn ,

by writing it out in long form.

S n = a + ar + ar 2 + ar 3 + ... + ar n ?1

By: Crystal Hull

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