If $p^{th}$, $q^{th}$, $r^{th}$ and $s^{th}$ terms of an $A.P$ are in $G.P$ then $p-q,q-r,r-s$ are in $G.P$.

As $p^{th},q^{th},r^{th}$ & $s^{th}$ terms are in G.P

$\therefore \frac{p}{q}=\frac{q}{r}=\frac{r}{s}$

OR $pr=q^2;r^2=qs$ & $ps=qr$

so, Lets Imagine $p-q,q-r$ & $r-s$ are in G.P.

so, $\frac{p-q}{q-r}=\frac{q-r}{r-s}$

OR $(p-q)(r-s)=(q-r{)}^2$

$pr-ps-qr+qs=q^2+r^2-2qr$

as $q^2=pr,r^2=qs$ & $ps=qr$

we get

$q^2-qr-qr+r^2=q^2+r^2-2qr$

$q^2+r^2-2qr=q^2+r^2-2qr$

L.H.S=R.H.S

$\therefore$ Our hypothesis is true.