[Math] How to find the common ratio of a geometric sequence

Question

A geometric sequence has its first term equal to $12$ and its fourth term equal to $-96$.

• How do I find the common ratio?
• And find the sum of the first $14$ terms

Answer 1

Terms of a geometric series are $a, ar, ar^2, ar^3, ...$,

where $a$ is the first term and $r$ is the common ratio.

In this case, $a=12$ and $ar^3=-96$, so $r^3=-8$, so $r=-2$.

The sum of the first $n$ terms of a geometric series (with $r\ne1$) is $a\dfrac{1-r^n}{1-r}$.

Can you take it from here?