If the $p^{th}$, $q^{th}$ and $r^{th}$ terms of a GP. are again in G.P., then which one of the following is correct?

Let R be the common ratio of this GP and a be the first term $p^{th}$ term is ${\mathrm{aR}}^{\mathrm{p}-1},$ $q^{th}$ term is ${\mathrm{aR}}^{q-1},$ and $r^{th}$ term is ${\mathrm{aR}}^{r-1},$ 

Since p,q and r are in G.P. then

${\left(a{R}^{q-1}\right)}^2=a{R}^{p-1}\cdot a{R}^{r-1}$

$\begin{array}{l}\Rightarrow {\mathrm{a}}^2{\mathrm{R}}^{2\mathrm{q}-2}={\mathrm{a}}^2{\mathrm{R}}^{\mathrm{p}+\mathrm{r}-2}\\ \Rightarrow {\mathrm{R}}^{2\mathrm{q}-2}={\mathrm{R}}^{\mathrm{p}+\mathrm{r}-2}\\ \Rightarrow 2\mathrm{q}-2=\mathrm{p}+\mathrm{r}-2\end{array}$

$\Rightarrow 2\mathrm{q}=\mathrm{p}+\mathrm{r}\Rightarrow \mathrm{p},\mathrm{q},\mathrm{r}$ are in A.P.