Question

If $p$th, $q$th, $r$th and $s$th terms of a G.p,prove that;
$(p-q)$, $(q-r)$ and $(r-s)$ are also in G.P.

Answer 1

Suppose a G.P is

$a,a{r}_1,a{r}_1^2.......a{r}_1^n\to$ First term =a Ratio = $r_1$

pth term = $a{r}_1^{p-1}$ qth term = $a{r}_1^{q-1}$

rth term = $a{r}_1^{r-1}$ sth term = $a{r}_1^{s-1}$

If they are in G.P. then

$\frac{a{r}_1^{q-1}}{a{r}_1^{p-1}}=\frac{a{r}_1^{r-1}}{a{r}_1^{q-1}}=\frac{a{r}_1^{s-1}}{a{r}_1^{r-1}}$

$(q-p)=(r-q)=(s-r)$

$(p-q)=(q-r)=(r-s)$

If they are equal then they are also in G.P. with ratio $1$.