Mathematics - Sequence & Series Question with Solution | TestHub

Given series is $$\begin{array}{rl}& \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\dots ..\\ & n^{\text{th }}\text{ term }=\frac{1}{\sqrt{n}+\sqrt{n+1}}\\ & \therefore 15^{\text{th }}\text{ term }=\frac{1}{\sqrt{15}+\sqrt{16}}\end{array}$$ Thus, given series upto 15 terms is $$\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\dots \dots +\frac{1}{\sqrt{15}+\sqrt{16}}$$ This can be re-written as $$\frac{1-\sqrt{2}}{-1}+\frac{\sqrt{2}-\sqrt{3}}{-1}+\frac{\sqrt{3}-\sqrt{4}}{-1}+\dots \dots +\frac{\sqrt{15}-\sqrt{16}}{-1}$$ (By rationalization) $$\begin{array}{c}=-1+\sqrt{2}-\sqrt{2}+\sqrt{3}-\sqrt{3}+\sqrt{4}+\dots ..-\sqrt{14}+\sqrt{15}\\ -\sqrt{15}+\sqrt{16}\\ =-1+\sqrt{16}=-1+4=3\end{array}$$ Hence, the required sum $=3$