Question

The sum of the series $1+2\cdot 2+3\cdot 2^2+4\cdot 2^3+5\cdot 2^4+\cdots +100\cdot 2^{99}$, is

• A. $99\cdot 2^{100}$
• B. $99\cdot 2^{100}+1$
• C. $100\cdot 2^{100}$
• D. none of these

Answer 1

The correct option is B $99\cdot 2^{100}+1$

Let $S=1+2\cdot 2+3\cdot 2^2+{4\cdot 2}^3+{5\cdot 2}^4+\cdots 100\cdot 2^{99}$ .................. (1)
Multiply (1) by $2$
$2S=2+2\cdot 2^2+{3\cdot 2}^3+\cdots 100\cdot 2^{100}$............(2)
Subtracting (1) and (2) gives
$(1-2)S=1+2+2^2+2^3+\cdots 2^{99}-100\cdot 2^{100}$
$\Rightarrow -S=\frac{2^{100}-1}{2-1}-100.2^{100}\Rightarrow -S=-1-99.2^{100}$
$\Rightarrow S=1+99\cdot 2^{100}$