Recently, I've stumbled upon an equation (9th grade) that I know nothing about. It looks like this:

${\frac {1} {\sqrt {5}+ \sqrt {2}}}+{\frac {1} {\sqrt {8}+ \sqrt {5}}} +{\frac {1} {\sqrt {11}+ \sqrt {8}}}+...+{\frac {1} {\sqrt {38}+ \sqrt {35}}}$

Obviously, if you continue by replacing three dots, you'll get:

${\frac {1} {\sqrt {14}+ \sqrt {11}}} + {\frac {1} {\sqrt {17}+ \sqrt {14}}}$ so on, so on until last.

But what am I supposed to do to solve this? Roughly translating from my language the task looks like: Evaluate the sum.

I mean, it doesn't look like arithmetic progression, because I am not able to find d or geometric progression to find q, so what should I do to solve it?


Hint: square roots in denominators are usually not nice to work with. So get rid of them: Expand the first fraction by $\sqrt5-\sqrt2$, the second fraction by $\sqrt8-\sqrt5$, and so on. Then add them together.

  • $\begingroup$ Helped out, thanks a lot! $\endgroup$ – Eugen Eray Apr 1 at 7:12

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