If  m is the A.M of the distinct real numbers  $\mathcal{l}$ and  $n,(\mathcal{l}n>1)$ and  $G_1,G_2,G_3$ are three G.M’s between  $\mathcal{l}$  and  $n$  then  $G_1^4+2G_2^4+G_3^4=\mathrm{.....}$ 

 $$\begin{gathered} 2m=\mathcal{l}+n \\ \mathcal{l},G_1,G_2,G_3,nareinGP \\ \frac{G_1}{\mathcal{l}}=\frac{G_2}{G_1}=\frac{G_3}{G_2}=\frac{n}{G_3} \\ {G}_1^2=\mathcal{l}{G}_2,G_2^2=G_1{G}_3,G_3^2=n{G}_2,G_1{G}_3=\mathcal{l}n \\ {G}_1^4+2G_2^4+G_3^4={\mathcal{l}}^2{G}_2^2+2G_2^4+n^2{G}_2^2 \\ =G_2^2({\mathcal{l}}^2+2\mathcal{l}n+n^2) \\ =\mathcal{l}n{(\mathcal{l}+n)}^2 \\ =4\mathcal{l}n{m}^2 \end{gathered}$$