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 $a, {A_1}, {A_2}, b$ are in $A.P.$,we have ${A_1} - a = b - {A_2}$,which implies ${A_1} + {A_2} = a +

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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 $a, {A_1}, {A_2}, b$ are in $A.P.$,we have ${A_1} - a = b - {A_2}$,which implies ${A_1} + {A_2} = a + b$ ......$(i)$Since $a, {G_1}, {G_2}, b$ are in $G.P.$,we have $\frac{{{G_1}}}{a} = \frac{b}{{{G_2}}}$,which implies ${G_1}{G_2} = ab$ ......$(ii)$Since $a, {H_1}, {H_2}, b$ are in $H.P.$,we have $\frac{1}{{{H_1}}} - \frac{1}{a} = \frac{1}{b} - \frac{1}{{{H_2}}}$,which implies $\frac{1}{{{H_1}}} + \frac{1}{{{H_2}}} = \frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}$.Substituting the values from $(i)$ and $(ii)$,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 $AM's$,$GM's$ and $HM's$ between two quantities,then the value of $\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}}$ is

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