Question

$1+3+7+15+31...nterms=$

• A. $2^{n+1}–n$
• B. $2^{n+1}–n–2$
• C. $2^n–n–2$
• D. $noneofthese$

Answer 1

The correct option is C

$2^n–n–2$

Explanation for the correct option:

Step 1. Rearrange given series to find the general term

Let $T_n$ is the $nth$ term

$\Rightarrow S=1+3+7+15+31+......+T_n$ …....(1)

$\Rightarrow S=1+3+7+15+......+T_{n-1}+T_n$ …....(2)

Subtracting equation (2) - (1), we get

$\Rightarrow 0=1+[2+4+8+16+.....(T_n-T_{n-1}\left)\right]-T_n$

$\Rightarrow T_n=1+2+4+8+\dots (T_n-T_{n-1})$

$=1+2(1+2^2+2^3+\dots nterms)$

$=2^n-1$

Step 2. Find the required sum:

$\Rightarrow S=\sum T_n=\sum \left(2^n-1\right)$

. $=\sum 2^n–\sum 1$

$=\frac{2(2^n-1)}{(2-1)}–n$ $\left[\mathbf{\because }\mathit{S}\mathit{u}\mathit{m}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{G}\mathbf{.}\mathit{P}\mathbf{.}\mathbf{=}\frac{\mathbf{a}\mathbf{(}{\mathbf{r}}^{\mathbf{n}}\mathbf{-}\mathbf{1}\mathbf{)}}{(r-1)}\right]$

$=2^{n+1}–2–n$

Hence, Option ‘C’ is Correct.