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Reciprocal Radical Sum

Problem

Consider the following series.

$$S(n) = \dfrac{1}{\sqrt{1} + \sqrt{2}} + \dfrac{1}{\sqrt{2} + \sqrt{3}} + \dfrac{1}{\sqrt{3} + \sqrt{4}} + ... + \dfrac{1}{\sqrt{n} + \sqrt{n+1}}$$

For which values of $n$ is $S(n)$ rational?

Problem ID: 266 (05 Feb 2006)     Difficulty: 4 Star

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