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

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Question

(a) Let $a$ and $b$ two numbers respectively.

Sum of $n$ $A.M.'s$ $ = n 	imes $single $A.M.$

$ \Rightarrow $${A_1} + {A_2} = 2 	imes \left

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Vedclass · Answer

(a) Let $a$ and $b$ two numbers respectively.

Sum of $n$ $A.M.'s$ $ = n 	imes $single $A.M.$

$ \Rightarrow $${A_1} + {A_2} = 2 	imes \left( {\frac{{a + b}}{2}} ight) = a + b$

Product of $n$ $G.M.'s$ = (Single $G.M.$)$^n$

$ \Rightarrow $ ${G_1}.{G_2} = {(\sqrt {ab} )^2} = ab$

$\frac{1}{a},\;\frac{1}{{{H_1}}},\;\frac{1}{{{H_2}}},\;\frac{1}{b}$ are in $A.P.$

$ \Rightarrow $ $\frac{1}{{{H_1}}} + \frac{1}{{{H_2}}} = \frac{1}{a} + \frac{1}{b} = \frac{{a + b}}{{ab}}$

$ \Rightarrow $$\frac{{{H_1}{H_2}}}{{{H_1} + {H_2}}} = \frac{{{G_1}{G_2}}}{{{A_1} + {A_2}}}$

$ \Rightarrow $ $\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}} 	imes \frac{{{H_1} + {H_2}}}{{{A_1} + {A_2}}} = 1$

If ${A_1},\;{A_2};{G_1},\;{G_2}$ and ${H_1},\;{H_2}$ be 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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