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Sum of n Terms

If A1,A2;G1...

Question

If $A_{1}, A_{2}; G_{1}, G_{2}; H_{1}, H_{2}$ be two $A . M .^{'} s, G . M .^{'} s$ and $H . M .^{'} s$ between two positive numbers then prove:
$\frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}}$ .

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Solution

Let the 2 numbers be $a, b$

$a, A_{1}, A_{2}, b$ are in A.P

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

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

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

$a, H_{1}, H_{2}, b$ are in H.P

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

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

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

From (1) and (2) we have,

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

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

Hence proved.

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