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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 ratioThe 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 



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