If $a_{1}, a_{2}, . . . a_{n}$ be an A.P. of +ive terms, then $n \sum k = 1 a_{k} \geq n \sqrt{a_{1}^{2} + (n - 1) d a_{1}}$ where d is common difference of A.P. If you think this is true write $1$ otherwise write $0$ ?

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Solution

$S_{n} = \frac{n}{2} [a_{1} + a_{n}] \geq n \sqrt{a_{1} . a_{n}} ∵ A . M . \geq G . M .$
or $S_{n} \geq n \sqrt{a_{1} [a_{1} + (n - 1) d]} = n \sqrt{a_{1}^{2} + (n - 1) d a_{1}}$
Thus answer is $1$ .

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