If {A_1}, {A_2}; {G_1}, {G_2} and {H_1}, {H_2 | vedclass

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(A) Let $a$ and $b$ be two numbers.The sum of $n$ $A.M.s$ is given by $n 	imes (	ext{single } A.M.)$,so ${A_1} + {A_2} = 2 	imes \left( \frac{a + b

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(A) Let $a$ and $b$ be two numbers.The sum of $n$ $A.M.s$ is given by $n 	imes (	ext{single } A.M.)$,so ${A_1} + {A_2} = 2 	imes \left( \frac{a + b}{2} ight) = a + b$.The product of $n$ $G.M.s$ is given by $(	ext{single } G.M.)^n$,so ${G_1}{G_2} = (\sqrt{ab})^2 = ab$.Since ${H_1}, {H_2}$ are $H.M.s$ between $a$ and $b$,the sequence $\frac{1}{a}, \frac{1}{H_1}, \frac{1}{H_2}, \frac{1}{b}$ is in $A.P$.Thus,$\frac{1}{H_1} + \frac{1}{H_2} = \frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}$.This simplifies to $\frac{H_1 + H_2}{H_1 H_2} = \frac{a + b}{ab}$.Substituting the values,we get $\frac{H_1 + H_2}{H_1 H_2} = \frac{A_1 + A_2}{G_1 G_2}$.Rearranging the terms,we have $\frac{G_1 G_2}{H_1 H_2} 	imes \frac{H_1 + H_2}{A_1 + A_2} = 1$.

If ${A_1}, {A_2}$; ${G_1}, {G_2}$ and ${H_1}, {H_2}$ are two $A.M.s$,$G.M.s$ and $H.M.s$ between two numbers respectively,then $\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}} \times \frac{{{H_1} + {H_2}}}{{{A_1} + {A_2}}} = $

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