Let $G_{1},G_{2}$ are two geometric means and A is the arithmetic mean between two positive numbers,

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Let $G_{1},G_{2}$ are two geometric means and A is the arithmetic mean between two positive numbers, then $\frac{G_{1}^{2}}{G_{2}} + \frac{G_{2}^{2}}{G_{1}} =$

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Sol. [Key Concept: Two G.M.s $G_{1,}G_{2}$ and A.M. $A$ between two numbers: $\frac{G_{1}^{2}}{G_{2}} + \frac{G_{2}^{2}}{G_{1}} = 2A$.]$a,G_{1},G_{2},b$ are in G.P.$$\Rightarrow \ G_{1} = a^{2/3}{	ext{ }b}^{1/3},G_{2} = a^{1/3}{	ext{ }b}^{2/3}$$$a,A,b$ are in A.P.$$\begin{matrix} \Rightarrow \ A & \ = \frac{a + b}{2} \ \frac{G_{1}^{3} + G_{2}^{3}}{G_{1}G_{2}} & \ = \frac{a^{2}	ext{ }b + {ab}^{2}}{ab} = a + b = 2	ext{ }A \end{matrix}$$

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