Simplify $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots+ \frac{1}{\sqrt{24} + \sqrt{25}}$ Simplify$$\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{24} + \sqrt{25}}.$$
I know you can solve this using generating functions but I'm not totally sure.

Hint: Multiply top and bottom of $\dfrac{1}{\sqrt{k+1}+\sqrt{k}}$ by $\sqrt{k+1}-\sqrt{k}$, and watch the house of cards collapse. 
