Question

The sum of $n$ terms of the series $1+3+7+15+31+...n$ terms is .

• A. $2^{(n+1)}-3-n$
• B. $2^{(n+1)}-2-n$
• C. $2^{(n+1)}+3+n$
• D. $2^{(n+1)}-2-n$

Answer 1

The correct option is B $2^{(n+1)}-2-n$

We see that the difference between each successive term is $2,4,8,16,...$

This is a GP.

$S_n=1+3+7+15+31+...+T_{n-1}+T_n$ ... (1)

$S_n=1+3+7+15+...+T_{n-1}+T_n$ ... (2)

[Write the $S_n$ in such a way that $1^{st}$ term of equation (2) comes under $2^{nd}$ term of equation (1)]

Subtracting equation (2) from equation (1.)

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

$\Rightarrow T_n=1+2+4+8+16+...n$ terms

$\Rightarrow T_n=\frac{1\times (2^n-1)}{2-1}$

Hence, sum of $n$ terms,

$S_n=\sum t_n={\sum }_{i=1}^n(2^i-1)$

$=\sum 2^n-\sum 1=2^{(n+1)}-2-n$