Mathematics - Sequence & Series Question with Solution | TestHub

$\begin{array}{rl}& \frac{1}{1\cdot (1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2\mathrm{d})}+\dots \dots \dots \dots ..\\ & \frac{1}{(1+9\mathrm{d})(1+10\mathrm{d})}=5\end{array}$ $\begin{array}{rl}& \frac{1}{d}\left[\frac{(1+d)-1}{1\cdot (1+d)}+\frac{(1+2d)-(1-d)}{(1+d)(1+2d)}\right]+\dots \dots \dots \dots ...\\ & \frac{(1+10d)-(1+9d)}{(1+9d)(1+10d)}=5\\ & \frac{1}{d}[\left(1-\frac{1}{1+d}\right)+\left(\frac{1}{1+d}-\frac{1}{1+2d}\right)+\dots \dots \dots \dots ..\\ & \left(\frac{1}{1+9d}-\frac{1}{1+10d}\right)]=5\\ & \frac{1}{d}\left[1-\frac{1}{(1+10d)}\right]=5\\ & \frac{10d}{1+10d}=5d\\ & 50d=5\end{array}$