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  4. Let a, b be two positive numbers, where a>b and 4( GM )=5( HM ) for the numbers, then a is equal to-

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Q. Let $a, b$ be two positive numbers, where $a>b$ and $4( GM )=5( HM )$ for the numbers, then a is equal to-

Sequences and Series

Solution:

$4( GM )=5( HM )$
$\Rightarrow 4 \sqrt{a b}=5\left(\frac{2 a b}{a+b}\right) $
$\Rightarrow 4(a+b)^2=25 a b $
$\Rightarrow 4 a^2-17 a b+4 b^2=0 $
$\Rightarrow(4 a-b)(a-4 b)=0$
$\Rightarrow a=4 b (a=b / 4$ neglecting)