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:
Each term of a GP after the first equals the product of the preceding term and the common ratio.
Some geometric progressions, with their common ratios given, are as follows:
For a GP whose first term is a and whose common ratio is r, the terms are
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:
The nth term of a GP is found by multiplying the first term by the n − 1 power of the common ratio.
Find the sixth term of a GP with first term 5 and common ratio 4.
Solution: We ...