Evaluate the sum \[\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \dotsb + \frac{1}{\sqrt{15} + \sqrt{16}}.\]

(You might want to use a calculator to get an estimate of the answer, but in order to get the exact answer you will have to do it by hand!)

Can you find a similar sum that evaluates to \(5\)?

Can you find a similar sum that evaluates to a number that is not an integer?