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Root squares everywhere

Algebra Level 3

$\large \dfrac{1}{\sqrt{1}+\sqrt{2}}+ \dfrac{1}{\sqrt{2}+ \sqrt{3}}+ \dfrac{1}{\sqrt{3}+\sqrt{4}}+ \cdots+ \dfrac{1}{\sqrt{99}+\sqrt{100}} = \, ?$

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by Paola Ramírez

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