Question

The sum of series $1\cdot 2+2\cdot 2^2+3\cdot 2^3+\dots +100\cdot 2^{100}$ is

• A. $101\cdot 2^{101}+2$
• B. $99\cdot 2^{101}+2$
• C. $199\cdot 2^{100}+2$
• D. $99\cdot 2^{100}+2$

Answer 1

The correct option is B $99\cdot 2^{101}+2$

Let us denote the given series by $S$.
Let $S=1\cdot 2+2\cdot 2^2+3\cdot 2^3+\dots +{100.2}^{100}\dots \left(1\right)$
$\Rightarrow 2S=1\cdot 2^2+2\cdot 2^3+\dots +99\cdot 2^{100}+100\cdot 2^{101}\dots \left(2\right)$
On subtracting, we get
$\Rightarrow -S=2+2^2+2^3+\dots +2^{100}-100\cdot 2^{101}$
$\Rightarrow -S=2\left(\frac{2^{100}-1}{2-1}\right)-100\cdot 2^{101}$
$\Rightarrow S=-2^{101}+2+100\cdot 2^{101}\Rightarrow S=2+99\cdot 2^{101}$