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# Inequality problem (TIFR 2013 problem 31)

Question:

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.

September 8, 2017

## Limit of a sequence (TIFR 2013 problem 32)

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