Question:
The sum $\sum_{k=1}^{20} k \frac{1}{2^{k}}$ is equal to-
Correct Option: , 2
Solution:
$S=\sum_{k=1}^{20} \frac{1}{2^{k}}$
$S=\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{3^{2}}+\ldots+\frac{20}{2^{20}}$
$S \times \frac{1}{2}=\frac{1}{2^{2}}+\frac{2}{2^{3}}+\ldots+\frac{19}{2^{20}}+\frac{20}{2^{21}}$
$\Rightarrow\left(1-\frac{1}{2}\right) \mathrm{S}=\frac{1}{2}+\frac{1}{2^{2}}+\ldots+\frac{1}{2^{20}}-\frac{20}{2^{21}}$
$\Rightarrow S=2-\frac{11}{2^{19}}$