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Inequality

If A1 , A2...

Question

If $A_{1}, A_{2}; G_{1}, G_{2}$ and $H_{1}, H_{2}$ be two A.M.s, G.M.s and H.M.s between two quantities a and b then
prove that $\frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}}$

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Solution

$S u m o f n A . M . s = n \times s i n g l e A . M$

$A_{1} + A_{2} = 2 (\frac{a + b}{2}) = a + b$

$P r o d u c t o f G . M . s = (s i n g l e G . M .)^{n}$

$G_{1} G_{2} = (\sqrt{a b})^{2} = a b$

$\frac{1}{a}, \frac{1}{H_{1}}, \frac{1}{H_{2}}, \frac{1}{b}$ are in A.P

$\frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{1}{a} + \frac{1}{b}$

$\frac{H_{1} + H_{2}}{H_{1} H_{2}} = \frac{a + b}{a b}$

$\frac{H 1 + H 2}{H 1 H 2} = \frac{A 1 + A 2}{G 1 G 2}$

$⟹ \frac{G 1 G 2}{H 1 H 2} = \frac{A 1 + A 2}{H 1 + H 2}$

Hence Proved.

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