Let ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{a}}_4 \text{and }{\mathrm{a}}_5$ be such that ${\mathrm{a}}_1,{\mathrm{a}}_2 \text{and }{\mathrm{a}}_3$ are in A.P ,${\mathrm{a}}_2,{\mathrm{a}}_3 \text{and }{\mathrm{a}}_4$ are in G.P.; and ${\mathrm{a}}_3,{\mathrm{a}}_4 \text{and }{\mathrm{a}}_5$ are in H.P. Then ${\mathrm{a}}_1,{\mathrm{a}}_3 \text{and }{\mathrm{a}}_5$ are in

We have,

$\begin{array}{rl} & {\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3\text{are in A.P.}\Rightarrow 2{\mathrm{a}}_2={\mathrm{a}}_1+{\mathrm{a}}_3----\left(\mathrm{i}\right)\\ & {\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{a}}_4\text{are in G.P.}\Rightarrow {{\mathrm{a}}_3}^2={\mathrm{a}}_2{\mathrm{a}}_4-----\left(\mathrm{ii}\right)\\ & {\mathrm{a}}_3,{\mathrm{a}}_4,{\mathrm{a}}_5\text{are in H.P.}\Rightarrow {\mathrm{a}}_4=\frac{2{\mathrm{a}}_3{\mathrm{a}}_5}{{\mathrm{a}}_3+{\mathrm{a}}_5}-----\left(\mathrm{iii}\right)\end{array}$

Putting ${\mathrm{a}}_2=\frac{{\mathrm{a}}_1+{\mathrm{a}}_3}{2}$ and ${\mathrm{a}}_4=\frac{2{\mathrm{a}}_3{\mathrm{a}}_5}{{\mathrm{a}}_3+{\mathrm{a}}_5}$ in (ii), we get

$\begin{array}{rl}{\mathrm{a}}_3^2& =\frac{{\mathrm{a}}_1+{\mathrm{a}}_3}{2}\times \frac{2{\mathrm{a}}_3{\mathrm{a}}_5}{{\mathrm{a}}_3+{\mathrm{a}}_5}\\ \Rightarrow {\mathrm{a}}_3^2& ={\mathrm{a}}_1{\mathrm{a}}_5\Rightarrow {\mathrm{a}}_1,{\mathrm{a}}_3,{\mathrm{a}}_5\text{are in G.P.}\end{array}$