Sequences and Series: Telescoping Series - Sum

$\small \begin{aligned} S & = \frac 1{\sqrt 4 + \sqrt 7} + \frac 1{\sqrt 7 + \sqrt{10}} + \cdots + \frac 1{\sqrt{397}+\sqrt{400}} \\ & = \frac {\sqrt 7 - \sqrt 4}{(\sqrt 7 + \sqrt 4)(\sqrt 7 - \sqrt 4)} + \frac {\sqrt{10} - \sqrt 7}{(\sqrt{10} + \sqrt 7)(\sqrt{10} - \sqrt 7)} + \cdots + \frac {\sqrt{400} - \sqrt{397}}{(\sqrt{400} + \sqrt{397})(\sqrt{400} - \sqrt{397})} \\ & = \frac {\sqrt 7 - \sqrt 4}{7-4} + \frac {\sqrt{10} - \sqrt 7}{10-7} + \frac {\sqrt{13}-\sqrt{10}}{13-10} + \cdots + \frac {\sqrt{400} - \sqrt{397}}{400-397} \\ & = \frac {\cancel{\sqrt 7} - \sqrt 4 + \cancel{\sqrt{10}} - \cancel{\sqrt 7} + \cancel{\sqrt{13}} - \cancel{\sqrt{10}} + \cdots + \sqrt{400} -\cancel{\sqrt{397}}}3 \\ & = \frac {\sqrt{400}-\sqrt 4}3 = \frac {20-2}3 = \frac {18}3 = \boxed 6 \end{aligned}$

Rationalizing every term of the given expression results in $\begin{aligned} &\frac{1}{\sqrt{4}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{10}}+\cdots+\frac{1}{\sqrt{397}+\sqrt{400}}\\ &=\frac{1}{\sqrt{7}+\sqrt{4}}\times\frac{\sqrt{7}-\sqrt{4}}{\sqrt{7}-\sqrt{4}}+\frac{1}{\sqrt{10}+\sqrt{7}}\times\frac{\sqrt{10}-\sqrt{7}}{\sqrt{10}-\sqrt{7}}+\cdots+\frac{1}{\sqrt{400}+\sqrt{397}}\times\frac{\sqrt{400}-\sqrt{397}}{\sqrt{400}-\sqrt{397}}\\ &=\frac{\sqrt{7}-\sqrt{4}}{3}+\frac{\sqrt{10}-\sqrt{7}}{3}+\cdots+\frac{\sqrt{400}-\sqrt{397}}{3}\\ &=\frac{\sqrt{400}-\sqrt{4}}{3}\\ &=6. \end{aligned}$