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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 