Since x is A.M

$\begin{array}{r}\Rightarrow x=\frac{y+z}{2}\\ \Rightarrow 2x=y+z\end{array}$…(1)

$\begin{array}{l}\Rightarrow \frac{g_1}{y}=\frac{g_2}{g_1}=\frac{z}{g_2}\\ \Rightarrow g_1^2=g_{2}y\\ \Rightarrow g_1^3=g_1{g}_{2}y ...\left(2\right)\end{array}$

Also, ${\mathrm{g}}_2^2={\mathrm{g}}_1\mathrm{z}$

$\begin{array}{l}{g}_2^3=g_1{g}_{2}z\\ \Rightarrow g_1^2{g}_2^2=g_1{g}_{2}yz\\ \Rightarrow yz=g_1{g}_2\end{array}$…(3)

Adding equation (2) and (3)

$\begin{array}{l}{\mathrm{g}}_1^3+{\mathrm{g}}_2^3={\mathrm{yg}}_1{\mathrm{g}}_2+{\mathrm{zg}}_1{\mathrm{g}}_2={\mathrm{g}}_1{\mathrm{g}}_2(\mathrm{y}+\mathrm{z})\\ =\mathrm{yz}\cdot 2\mathrm{x}=2\mathrm{xyz}\end{array}$