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.