Question

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

Answer 1

Given : $p^{th},q^{th},r^{th},s^{th}$ are in A.P.

With the usual notation we have
$\frac{a+(p-1)d}{a+(q-1)d}=\frac{a+(q-1)d}{a+(r-1)d}=\frac{a+(r-1)d}{a+(s-1)d}$;
$\therefore$ each of these ratios
$=\frac{\{a+(p-1\left)d\right\}-\{a+(q-1\left)d\right\}}{\{a+(q-1\left)d\right\}-\{a+(r-1\left)d\right\}}=\frac{\{a+(q-1\left)d\right\}-\{a+(r-1\left)d\right\}}{\{a+(r-1\left)d\right\}-\{a+(s-1\left)d\right\}}$
$\Rightarrow \frac{p-q}{q-r}=\frac{q-r}{r-s}$.
Hence $p-q,q-r,r-s$ are in G.P.