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

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(A) Let the two quantities be $a$ and $b$.Since ${A_1}, {A_2}$ are $AMs$ between $a$ and $b$,$a, {A_1}, {A_2}, b$ are in $A.P.$Thus,${A_1} - a = b - {

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$\frac{{A_1 + A_2}}{{H_1 + H_2}}$

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(A) Let the two quantities be $a$ and $b$.Since ${A_1}, {A_2}$ are $AMs$ between $a$ and $b$,$a, {A_1}, {A_2}, b$ are in $A.P.$Thus,${A_1} - a = b - {A_2} \Rightarrow {A_1} + {A_2} = a + b$ ......$(i)$Since ${G_1}, {G_2}$ are $GMs$ between $a$ and $b$,$a, {G_1}, {G_2}, b$ are in $G.P.$Thus,$\frac{{G_1}}{a} = \frac{b}{{G_2}} \Rightarrow {G_1}{G_2} = ab$ ......$(ii)$Since ${H_1}, {H_2}$ are $HMs$ between $a$ and $b$,$a, {H_1}, {H_2}, b$ are in $H.P.$Thus,$\frac{1}{{H_1}} - \frac{1}{a} = \frac{1}{b} - \frac{1}{{H_2}} \Rightarrow \frac{1}{{H_1}} + \frac{1}{{H_2}} = \frac{1}{a} + \frac{1}{b}$$\Rightarrow \frac{{H_1 + H_2}}{{H_1 H_2}} = \frac{{a + b}}{{ab}}$Substituting $(i)$ and $(ii)$ into the expression,we get $\frac{{H_1 + H_2}}{{H_1 H_2}} = \frac{{A_1 + A_2}}{{G_1 G_2}}$Therefore,$\frac{{G_1 G_2}}{{H_1 H_2}} = \frac{{A_1 + A_2}}{{H_1 + H_2}}$.

If ${A_1}, {A_2}$; ${G_1}, {G_2}$ and ${H_1}, {H_2}$ are $AMs$,$GMs$,and $HMs$ between two quantities,then the value of $\frac{{G_1 G_2}}{{H_1 H_2}}$ is

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