Special Series in Sequences and Series

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Question

Suppose A.M. of two positive numbers be 7 and G.M. between them be 5. The A.M. between their squares is

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Solution

Let two positive numbers be a, b. Then
$a + b = 2 (7) = 14$ and $\sqrt{a b} = 5$ or $a b = 25$
Now, $a^{2} + b^{2} = (a + b)^{2} - 2 a b = 196 - 50 = 146$
$\Rightarrow \frac{1}{2} (a^{2} + b^{2}) = 73$

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