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Playing with Square Root

The sum

121+12+132+23+143+34++110099+99100\displaystyle \frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+\dots+\frac{1}{100\sqrt{99}+99\sqrt{100}}

can be expressed as ab,\frac{a}{b}, where aa and bb are coprime positive integers.

What is the value of a+ba+b?

This problem is adapted from a past year KMC Contest.


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