Mathematics - Sequence & Series Question with Solution | TestHub

$\begin{array}{rl}& \frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\dots +\frac{1}{\sqrt{99}+\sqrt{100}}=\mathrm{m}\\ & \frac{\sqrt{1}-\sqrt{2}}{-1}+\frac{\sqrt{2}-\sqrt{3}}{-1}\dots \frac{\sqrt{99}-\sqrt{100}}{-1}=\mathrm{m}\end{array}$ $\begin{array}{rl}& \sqrt{100}-1=\mathrm{m}\Rightarrow \mathrm{m}=9\\ & \frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\dots \frac{1}{99\cdot 100}=\mathrm{n}\\ & \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}\dots \frac{1}{99}-\frac{1}{100}=\mathrm{n}\\ & 1-\frac{1}{100}=\mathrm{n}\\ & \frac{99}{100}=\mathrm{n}\\ & (\mathrm{m},\mathrm{n})=\left(9,\frac{99}{100}\right)\\ & \Rightarrow 11(9)-100\left(\frac{99}{100}\right)\\ & =99-99=0\end{array}$ Ans.$11x-100y=0$