A **geometric sequence** is a sequence in which each term is found by multiplying the preceding term by the same value. Its general term is

*a ** _{n}*=

*a*

_{1}

*r*

^{n}^{ – 1 }

The value *r* is called the **common ratio.** It is found by taking any term in the sequence and dividing it by its preceding term.

Example 1

Find the common ratio in each of the following geometric sequences. Then express each sequence in the form *a ** _{n}*=

*a*

_{1}

*r*

^{n}^{ – 1 }and find the eighth term of the sequence.

- 1, 3, 9, 27, …
- 64, –16, 4, –1, …
- 16, 24, 36, 54, …
- 1, 3, 9, 27, …
- Since
- Then
*a*= 1(3_{n}^{n}^{–1 }) Therefore, the eighth term of the sequence is - 64, –16, 4, –1, …
- Since
- Then
- Therefore, the eighth term of the sequence is
- 16, 24, 36, 54, …
- Since
- Then
- Therefore, the eighth term of the sequence is