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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}}21+121+32+231+43+341+⋯+10099+991001
can be expressed as ab,\frac{a}{b},ba, where aaa and bbb are coprime positive integers.
What is the value of a+ba+ba+b?
This problem is adapted from a past year KMC Contest.
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