Find the sum of $\frac{1}{\sqrt1+\sqrt3} + \frac{1}{\sqrt3+\sqrt5} + \frac{1}{\sqrt5+\sqrt7} + ... \frac{1}{\sqrt79+\sqrt81}$ Find the sum of $\frac{1}{\sqrt1+\sqrt3} + \frac{1}{\sqrt3+\sqrt5} + \frac{1}{\sqrt5+\sqrt7} + ... \frac{1}{\sqrt{79}+\sqrt{81}}$
I've thought about multiplying every fraction by 1, but like this $\frac{1} {\sqrt1+\sqrt3} *\frac{\sqrt1+\sqrt3}{\sqrt1+\sqrt3}$, $\frac{1} {\sqrt3+\sqrt5} *\frac{\sqrt3+\sqrt5}{\sqrt3+\sqrt5}...$
After multiplying I end up with $\frac{\sqrt1+\sqrt3} {(\sqrt1+\sqrt3)^2} + \frac{\sqrt3+\sqrt5} {(\sqrt3+\sqrt5)^2}  ...$ but I don't have any more ideas than this, am I on the right path?
I apologize for the question's simplicity, but I'm a younger math enthusiast so questions like this are harder for me.

Hint: $\frac 1{\sqrt{n} + \sqrt{n+2}} \cdot \frac {\sqrt{n+2} - \sqrt{n}}{\sqrt{n+2} - \sqrt{n}} = \frac{\sqrt{n+2} - \sqrt{n}}{2}$
