What is the value of<br>\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\dots+\frac{1}{\sqrt{99}+\sqrt{100}}?

Explanation

Rationalizing each term gives \sqrt{k+1} - \sqrt{k}. The sum forms a telescoping series: (\sqrt{2}-1) + (\sqrt{3}-\sqrt{2}) + \dots + (\sqrt{100}-\sqrt{99}). All intermediate terms cancel out, leaving \sqrt{100} - 1 = 10 - 1 = 9.

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