If between any two quantities there be inserted two arithmetic means $A_{1},{ ext{ }A}_{2}$; two ge

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If between any two quantities there be inserted two arithmetic means $A_{1},{	ext{ }A}_{2}$; two geometric means $G_{1},G_{2}$; and two harmonic means $H_{1},H_{2}$, then $G_{1}G_{2}:A_{1} + A_{2}$ is equal to

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Sol. [Key Concept: Insert 2 A.M.s, 2 G.M.s, 2 H.M.s between two numbers and combine the standard mean relations.]$${\frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{1}{a} + \frac{1}{	ext{ }b} = \frac{a + b}{ab} = \frac{A_{1} + A_{2}}{G_{1}G_{2}} }{\Rightarrow \frac{H_{1} + H_{2}}{H_{1}H_{2}} = \frac{A_{1} + A_{2}}{G_{1}G_{2}}}$$

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