Mathematics - Sequence & Series Question with Solution | TestHub

Let $S=1+2.2+3.2^2+4.2^2+\dots .+100.2^{99}$ It is an arithmetico - geometric series. On multiplying Eq. (i) by 2 and then subtracting it from Eq. (i), we get $S=1+2.2+3.2^2+4.2^3+\dots +100.2^{99}$ $2S=1.2+2.2^2\pm 3.2^3+\dots +99.2^{99}+100.2^{100}$ $\begin{array}{rl}-S& =1+2+2^2+2^3\dots +2^{99}-100.2^{100}\\ & \Rightarrow -S=\frac{1\left(2^{100}-1\right)}{2-1}-100.2^{100}\\ & \Rightarrow -S=2^{100}-1-100.2^{100}\\ & \Rightarrow -S=-1-99.2^{100}\\ & \Rightarrow S=99.2^{100}+1\end{array}$