A.M. of the square root of products taken two together of n +ive quantities is less than A.M. of the n quantities itself.

True

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False

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Solution

$\sqrt{(a_{1} a_{2})} \leq \frac{a_{1} + a_{2}}{2}, \sqrt{(a_{1} a_{3})} \leq \frac{a_{1} + a_{3}}{2} \dots \dots$
There will be $^{n} C_{2} = \frac{n (n - 1)}{2}$ inequalities of this type.
In the R.H.S. $= a_{1}$ will be (n-1) times and so also $a_{2}, a_{3}$ etc. Adding all these inequalities we get
${\sqrt{(a_{1} a_{2})} + \sqrt{(a_{1} a_{3})} + \dots \dots + \frac{n (n - 1)}{2}} < (n - 1) {\frac{a_{1} + a_{2} + \dots \dots + a_{n}}{2}} \dots \dots (1)$

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