JEE Novice - (18)

$\begin{aligned} \sum_{k=1}^{10^6} {\frac{1}{\sqrt{k}+\sqrt{k+1}}} & = \sum_{k=1}^{10^6} {\frac{\sqrt{k+1}-\sqrt{k}}{\left(\sqrt{k+1}+\sqrt{k}\right) \left(\sqrt{k+1}-\sqrt{k}\right)}} \\ & = \sum_{k=1}^{10^6} {\left( \sqrt{k+1}-\sqrt{k}\right)} \\ & = \sum_{k=2}^{10^6+1} {\sqrt{k}} - \sum_{k=1}^{10^6} {\sqrt{k}} \\ & = \sqrt{10^6+1} - \sqrt{1} \end{aligned}$

$\Rightarrow a + b = 1000001+1 = \boxed{1000002}$