Suppose $(m+n)$th term of a $G.P.$ is $p$ and $(m – n)$th term is $q,$ then its $n$th is

Let a be the first term and r be the common ratio.

We have

$\begin{array}{l} p=a_{m+n}=a{r}^{m+n-1}\\ q=a_{m-n}=a{r}^{m-n-1}\\ \Rightarrow \frac{p}{q}=\frac{a_{m+n}}{a_{m-n}}=r^{2n}\Rightarrow r={\left(\frac{p}{q}\right)}^{1/2n}\end{array}$

Now, $a_n=a{r}^{n-1}=\frac{a{r}^{m+n-1}}{r^m}$

$=p{\left(\frac{p}{q}\right)}^{-m/2n}=p{\left(\frac{q}{p}\right)}^{m/2n}$