If p, q, r, s are in G. P., show that p + q, q + r, r + s are also in G.P.
Answer$p, q, r, s$ are in G.P.
$
\begin{array}{ll}
\therefore \quad & \frac{\mathrm{q}}{\mathrm{p}}=\frac{\mathrm{r}}{\mathrm{q}}=\frac{\mathrm{s}}{\mathrm{r}} \\
& \text { Let } \frac{\mathrm{q}}{\mathrm{p}}=\frac{\mathrm{r}}{\mathrm{q}}=\frac{\mathrm{s}}{\mathrm{r}}=\mathrm{k} \\
\therefore & \mathrm{q}=\mathrm{pk}, \mathrm{r}=\mathrm{qk}, \mathrm{s}=\mathrm{rk}
\end{array}
$
We have to prove that $\mathrm{p}+\mathrm{q}, \mathrm{q}+\mathrm{r}, \mathrm{r}+\mathrm{s}$ are in G.P. i.e., to prove that $\frac{q+r}{p+q}=\frac{r+s}{q+r}$
$
\begin{aligned}
& \text { L.H.S. }=\frac{q+r}{p+q}=\frac{q+q k}{p+p k}=\frac{q(1+k)}{p(1+k)}=\frac{q}{p}=k \\
& \text { R.H.S. }=\frac{r+s}{q+r}=\frac{r+r k}{q+q k}=\frac{r(1+k)}{q(1+k)}=\frac{r}{q}=k \\
& \therefore \quad \frac{q+r}{p+q}=\frac{r+s}{q+r} \\
& \therefore p+q, q+r, r+s \text { are in G.P. }
\end{aligned}
$