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.

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