Math

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