# How do I find the equation of a geometric sequence?

Apr 25, 2015

A geometric sequence is a list of terms that have a common ratio between them. For example, 4, 8, 16, 32, 64, 128, .... has a ratio of 2. For example, 100, 50, 25, 12.5, ... has a ratio of $\frac{1}{2}$. The formula to compute the $n t h$ term of a geometric sequence is: ${a}_{n} = {a}_{1} \cdot {r}^{n - 1}$. The variable $r$ stands for the ratio, ${a}_{1}$ stands for the first term in the sequence, and $n$ stands for term numbers. (always counting numbers)

Let's say you are given the following list of terms and asked to write the "equation" for it:

$8 , \frac{8}{3} , \frac{8}{9} , \frac{8}{27} , \ldots$

What is the first term? 8
What is the ratio? $\frac{1}{3}$

So, ${a}_{n} = 8 \cdot {\left(\frac{1}{3}\right)}^{n - 1}$ will generate any term you want!