Select Page

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.