ABC26GN1192 · Simplification

Required sum $=\frac{1}{2}+\left(\frac{1}{3}+\frac{2}{3}\right)+\left(\frac{1}{4}+\frac{2}{4}+\frac{3}{4}\right)+\left(\frac{1}{5}+\frac{2}{5}+\frac{3}{5}+\frac{4}{5}\right)+\ldots$ $$\begin{aligned} +\left(\frac{1}{100}+\frac{2}{100}+\right. & \left.\frac{3}{100}+\ldots+\frac{99}{100}\right) \\ & =\frac{1}{2}+\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+\ldots . .+\frac{99}{2} \\ & =\frac{1}{2} \times(1+2+3+\ldots . .+99) \\ & =\frac{1}{2} \times \frac{99 \times 100}{2}=2475 . \end{aligned}$$