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 $

False

Discussion:

$ \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 $n$.

Some Useful Links:

• Self learning course for TIFR
• Cheenta YouTube Channel
• Cyclic Groups - TIFR Entrance - Video