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AM,GM,HM Inequality

If between an...

Question

If between any two quantities there be inserted two arithmetic means $A_{1}, A_{2}$ ; two geometric means $G_{1}, G_{2} : H_{1} H_{2} = A_{1} + A_{2} : H_{1} + H_{2}$ .

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Solution

Let the quantities be $a$ and $b$ . Then, $a, A_{1}, A_{2}, b$ are in $a . p .$ Thus, we can write

$A_{1} - a = b - A_{2}$ or $A_{1} + A_{2} = a + b ⟶ (I)$

If $a, G_{1}, G_{2}$ are in $G . P .$ Then we can write

$\frac{G_{1}}{a} = \frac{b}{G_{2}}$ or $G_{1} G_{2} = a b ⟶ (I I)$

If $a, H_{1}, H_{2}, b$ are in $H P$ . Then $\frac{1}{a}, \frac{1}{H_{1}}, \frac{1}{H_{2}}, \frac{1}{b}$ will be in $A . P .$

$∴ \frac{1}{H_{1}} - \frac{1}{a} = \frac{1}{b} - \frac{1}{H_{2}}$ or,

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

From $(I)$ and $(I I),$

$\frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{A_{1} + A_{2}}{G_{1} G_{2}}$

$⟹ \frac{H_{1} + H_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{G_{1} G_{2}}$

$⟹ \frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}}$

$∴ G_{1} G_{2} : H_{1} H_{2} = (A_{1} + A_{2}) : (H_{1} + H_{2})$

Hence proved

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