Technical Mathematics, Sixth Edition

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 a_n 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: 2, 4, 8, 16, ... (r = 2) 27, 9, 3, 1, , ... () −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 a_n 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 ...