If A_1, A_2 are two arithmetic means,G_1, G_2 | vedclass

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(C) Let the two numbers be $a$ and $b$.For two arithmetic means $A_1, A_2$ between $a$ and $b$,the sum is $A_1 + A_2 = 2 	imes \frac{a+b}{2} = a+b$.F

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(C) Let the two numbers be $a$ and $b$.For two arithmetic means $A_1, A_2$ between $a$ and $b$,the sum is $A_1 + A_2 = 2 	imes \frac{a+b}{2} = a+b$.For two geometric means $G_1, G_2$ between $a$ and $b$,the product is $G_1 G_2 = ab$.For two harmonic means $H_1, H_2$ between $a$ and $b$,the sum of their reciprocals is $\frac{1}{H_1} + \frac{1}{H_2} = 2 	imes \frac{\frac{1}{a} + \frac{1}{b}}{2} = \frac{a+b}{ab}$.This simplifies to $\frac{H_1 + H_2}{H_1 H_2} = \frac{a+b}{ab}$.Substituting $a+b = A_1 + A_2$ and $ab = G_1 G_2$,we get $\frac{H_1 + H_2}{H_1 H_2} = \frac{A_1 + A_2}{G_1 G_2}$.Rearranging the terms,we get $\frac{A_1 + A_2}{H_1 + H_2} \cdot \frac{H_1 H_2}{G_1 G_2} = 1$.

If $A_1, A_2$ are two arithmetic means,$G_1, G_2$ are two geometric means,and $H_1, H_2$ are two harmonic means between two numbers,then $\frac{A_1 + A_2}{H_1 + H_2} \cdot \frac{H_1 H_2}{G_1 G_2} = \dots$

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