Find the value of $\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \dots + \frac{1}{\sqrt{16} + \sqrt{15}}$.

Explanation

1. 01

   Rationalise one term: $\frac{1}{\sqrt{n + 1} + \sqrt{n}} = \frac{\sqrt{n + 1} - \sqrt{n}}{(n + 1) - n} = \sqrt{n + 1} - \sqrt{n}$.

2. 02

   The sum becomes $(\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2}) + \dots + (\sqrt{16} - \sqrt{15})$.

3. 03

   All middle terms cancel, leaving $\sqrt{16} - 1 = 4 - 1 = 3$.