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**Geometric Sequences**

A **geometric sequence** has the following general form:

##### In a geometric sequence each term is multiplied or divided by a constant number to get the next term. This constant,, is called the **common ratio**. The general term of an arithmetic sequence is given by:

.

##### A geometric sequence is an increasing sequence if the terms are multiplied by a number greater than one to get the next terms. A decreasing geometric sequence can be formed by multiplying each term by a number less than one to get the next terms.

**Example: **Is the following sequence an increasing or a decreasing geometric sequence?

##### We find the common ratio as follows:

##### since the common ratio is less than one the sequence is a decreasing geometric sequence.

**Example: **For a geometric sequence, and Write the first four terms of the sequence.

##### We can use the definition of the common ratio to find the fourth term of the sequence:

##### Using the same approach, we can find the first three term of the sequence:

**Example: **For a geometric sequence, and . Find the second term of the sequence.

##### For we get , and for we get