The fourth, seventh and tenth terms of $a$ G.P. are $p, q, r$ respectively, then :

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

Fourth term of G.P. : $p=T_4=a{r}^3 \left(i\right)$

Seventh term of G.P. :$q=T_7=a{r}^6 \dots \text{ (ii) }$

Tenth term of G.P. : $r=T_{10}=a{r}^9 ....\left(iii\right)$

Equ. (i) x Equ. (iii): 

$\begin{array}{l}pr=a{r}^3\times a{r}^9\Rightarrow pr=a^2{r}^{12}\Rightarrow pr={\left(a{r}^6\right)}^2\\ \Rightarrow pr=q^2\end{array}$