13.3 Geometric sequences  (Page 2/6)

The first four terms are $\left\{\mathrm{–2}\text{, }\mathrm{–8}\text{, }\mathrm{–32}\text{, }\mathrm{–128}\right\}.$

Using recursive formulas for geometric sequences

A recursive formula    allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

Using explicit formulas for geometric sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.

Let’s take a look at the sequence $\left\{18\text{, }36\text{, }72\text{, }144\text{, }288\text{, }\mathrm{...}\right\}.$ This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is

The graph of the sequence is shown in [link] .