Let $\mathrm{A}$ be the 1^st term and $\mathrm{R}$ be the common ratio of the given G.P.

Sum of the first $\mathrm{n}$ terms of the G.P. is $\mathrm{x}\mathrm{ }=\frac{\mathrm{A}(1-{\mathrm{R}}^{\mathrm{n}})}{1-\mathrm{R}}$. The next $\mathrm{n}$ terms form a G.P. series of $\mathrm{n}$ terms with the first term as the ${\left(\mathrm{n}+1\right)}^{\mathrm{t}\mathrm{h}}$ term of the given G.P. is given by ${\mathrm{t}}_{\mathrm{n}+1}=\mathrm{A}{\mathrm{R}}^{\mathrm{n}}$ . Sum of the next $\mathrm{n}$ terms of the G.P. is $\mathrm{y}\mathrm{ }\Rightarrow \mathrm{}\mathrm{y}\mathrm{ }=\frac{\mathrm{A}{\mathrm{R}}^{\mathrm{n}}(1-{\mathrm{R}}^{\mathrm{n}})}{1-\mathrm{R}}$ and 

$\mathrm{z}=\mathrm{A}{\mathrm{R}}^{2\mathrm{n}}\frac{\left(1-{\mathrm{R}}^{\mathrm{n}}\right)}{1-\mathrm{R}}$ $\Rightarrow \mathrm{ }{\mathrm{y}}^2=\mathrm{z}\mathrm{x}$

$\therefore$ $\mathrm{x}$, $\mathrm{y}$, $\mathrm{z}$ are in G.P.