Question

If the $p^{th}$, $q^{th}$, $r^{th}$ terms of an $A.P$ are in $G.P$ show that common ratio of the $G.P$ is $\frac{q-r}{p-q}$.

Answer 1

We are given that $T_p$, $T_q$ and $T_r$ of an $A.P.$ are in $G.P.$
$\therefore \frac{T_q}{T_p}=\frac{T_r}{T_q}=R=\frac{T_q-T_r}{T_p-T_q}$
(Ratio Prop.)
$\therefore R=\frac{[a+(q-1)d]-[a+(r-1)d]}{[a+(p-1)d]-[a+(q-1)d]}$
$=\frac{(q-r)d}{(p-q)d}=\frac{q-r}{p-q}$