$\text{If }{{\alpha }^{2}}+{{\beta }^{2}},\alpha \beta +\beta \gamma \text{ and }{{\beta }^{2}}+{{\gamma }^{2}}\text{ are in G}\text{.P, Prove that }\alpha \text{,}\beta ,\gamma \text{ are also in G}\text{.P}$
$\begin{align} & \text{The common ratio is } \\ & \frac{\alpha \beta +\beta \gamma }{{{\alpha }^{2}}+{{\beta }^{2}}}=\frac{{{\beta }^{2}}+{{\gamma }^{2}}}{\alpha \beta +\beta \gamma } \\ & {{(\alpha \beta +\beta \gamma )}^{2}}=({{\alpha }^{2}}+{{\beta }^{2}})({{\beta }^{2}}+{{\gamma }^{2}}) \\ & {{(\alpha \beta +\beta \gamma )}^{2}}={{(\alpha \beta )}^{2}}+{{(\alpha \gamma )}^{2}}+{{({{\beta }^{2}})}^{2}}+{{(\beta \gamma )}^{2}} \\ & {{(\alpha \beta +\beta \gamma )}^{2}}=\underbrace{{{(\alpha \beta )}^{2}}+{{(\beta \gamma )}^{2}}}_{{{(\alpha \beta +\beta \gamma )}^{2}}-2\alpha \gamma {{\beta }^{2}}}+{{(\alpha \gamma )}^{2}}+{{({{\beta }^{2}})}^{2}} \\ & {{(\alpha \beta +\beta \gamma )}^{2}}={{(\alpha \beta +\beta \gamma )}^{2}}-2\alpha \gamma {{\beta }^{2}}+{{(\alpha \gamma )}^{2}}+{{({{\beta }^{2}})}^{2}} \\ & {{({{\beta }^{2}})}^{2}}-2\alpha \gamma {{\beta }^{2}}+{{(\alpha \gamma )}^{2}}=0 \\ & {{({{\beta }^{2}}-\alpha \gamma )}^{2}}=0 \\ & {{\beta }^{2}}=\alpha \gamma ----(proved) \\ & \alpha ,\beta ,\gamma \text{ are also in a G}\text{.P}\text{.} \\\end{align}$