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Let’s discuss a problem from TIFR 2013 Problem 31 based on inequality.

Question: TIFR 2013 problem 31

True/False?

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

Hint:

Simplify the given inequality

Discussion:

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

$= \frac {1}{ \sqrt{n+1}+ \sqrt{n}} < \frac {1}{ \sqrt{n}}$ This holds for any natural number $n$.

So the inequality is actually true for all natural numbers.