# A Geometric Sequence has 3rd term of 12 and 5th term of 48. What are the possible values for the common ratio?

Sep 3, 2017

$\pm 2$

#### Explanation:

Suppose the first term of the GP is $a$, and the common ratio is $r$, then the GP sequence would be :

$\left\{a , a r , a {r}^{2} , a {r}^{3} , \ldots , a {r}^{n - 1}\right\}$

We are given that:

${3}^{r d} \text{ term} = 12 \implies a {r}^{2} = 12$ ..... [A]
${5}^{t h} \text{ term} = 48 \implies a {r}^{4} = 48$ ..... [B]

Eq [B] $\div i \mathrm{de}$ Eq [A]:

$\frac{a {r}^{4}}{a {r}^{2}} = \frac{48}{12}$
$\therefore {r}^{2} = 4$
$\therefore r = \pm 2$