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.

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