How do you find the common ratio of a geometric sequence if not given the first term?

Question

The only given values are the sum of an infinite geometric series which is equal to 9/2, and the second term which is equal to -2. How do I find the common ratio here?

Answer 1

Assume the geometric sequence to be

$$a,ar,ar^2...$$

(where $r$ is the common ratio)

Given that $ar=-2$ and $\frac{a}{1-r}=\frac92$

Two equations and two variables. I bet you can solve it now.

Answer 2

Here is a neater version. From the formula:

$$a = \frac{9}{2}(1-r)$$ $$ar = -2= \frac{9}{2}(1-r)r$$

and remember that $|r| < 1$ for the series to converge.