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### 20.3. Geometric Progressions

#### 20.3.1. Recursion Formula

A geometric sequence or geometric progression (GP) is one in which each term after the first is formed by multiplying the preceding term by a factor r, called the common ratio. Thus if an is any term of a GP, the recursion relation is as follows:

NOTE

Each term of a GP after the first equals the product of the preceding term and the common ratio.

## Example 25:

Some geometric progressions, with their common ratios given, are as follows:

1. 2, 4, 8, 16, ... (r = 2)

2. 27, 9, 3, 1, , ... ()

3. −1, 3, −9, 27, ... (r = −3)

#### 20.3.2. General Term

For a GP whose first term is a and whose common ratio is r, the terms are

a, ar, ar2, ar3, ar4, ...

We see that each term after the first is the product of the first term and a power of r, where the power of r is one less than the number n of the term. So the nth term an is given by the following equation:

NOTE

The nth term of a GP is found by multiplying the first term by the n − 1 power of the common ratio.

## Example 26:

Find the sixth term of a GP with first term 5 and common ratio 4.

Solution: We ...

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