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Inequality of square root function

This post contains a problem from TIFR 2013 Math paper D based on Inequality of square root function.

The inequality $ \sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n } $ is false for all in n such that $ 101 \le n \le 2000 $



$ \sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n } $

By cross multiplying we have  $ \sqrt {(n+1)n} - n < 1 $. That is $ \sqrt {n(n+1)} < (n+1) $  or $ n(n+1) < (n+1) ^2 $ or  n < n+1

This is true for all n.

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