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.
