If one A.M., A and two geometric means

Let the two positive numbers be $a$ and $b$.

$a, A$ and $b$ are in A.P.

$\therefore 2 A=a+b$    ....(i)

Also, $a, G_{1}, G_{2}$ and $b$ are in G.P.

$\therefore r=\left(\frac{b}{a}\right)^{\frac{1}{3}}$

Also, $G_{1}=a r$ and $G_{2}=a r^{2}$    ...(ii)

Now, LHS $=\frac{G_{1}{ }^{2}}{G_{2}}+\frac{G_{2}{ }^{2}}{G_{1}}$

$=\frac{(a r)^{2}}{a r^{2}}+\frac{\left(a r^{2}\right)^{2}}{a r} \quad[$ Using (ii) $]$

$=a+a r^{3}$

$=a+a\left(\left(\frac{b}{a}\right)^{\frac{1}{3}}\right)^{3}$

$=a+a\left(\frac{b}{a}\right)$

$=a+b$

$=2 A$

$=\mathrm{RHS} \quad[U \operatorname{sing}(\mathrm{i})]$