A Radical Reciprocation

Algebra Level 4

\[\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} +\cdots + \frac{1}{\sqrt{n}+\sqrt{n+1}} \]

Find the sum of all values of \(n\), \(1 \leq n \leq 100\), for which the expression above is rational.

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