If $A_{1}, A_{2}$ are two AM’s between two numbers $a$ and $b$ and $G_{1}, G_{2}$ be two GM’s between same two numbers, then $\frac{(A_{1} + A_{2})}{G_{1} . G_{2}} =$

$(a + b) / a b$

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$(a + b) / 2 a b$

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$2 a b / (a + b)$

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$a b / (a + b)$

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Solution

The correct option is A
$(a + b) / a b$

Explanation for the correct option:
Find the value of $\frac{(A_{1} + A_{2})}{G_{1} . G_{2}}$ :
Let $a, A_{1}, A_{2}, b$ are in AP.
Then $\frac{(a + b)}{2} = \frac{(A_{1} + A_{2})}{2}$
$\Rightarrow (A_{1} + A_{2}) = (a + b) . . . . . . . . . . . (i)$
Let $a, G_{1}, G_{2}, b$ are in GP.
$\Rightarrow \sqrt{a b} = \sqrt{G_{1} G_{2}} \Rightarrow G_{1} G_{2} = a b . . . . . . . . . . (i i)$
From equation (i) and (ii). we get
$\frac{(A_{1} + A_{2})}{G_{1} . G_{2}} = \frac{(a + b)}{a b}$
Hence, the correct option is ( A).

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