Question

The sum $1+3+7+15+31+..$.to n terms is

• A. $2^n-2-n$
• B. $2^{n-1}-2-n$
• C. $2^{n+1}-2-n$
• D. None of these

Answer 1

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

Given that,

$1+3+15+31+.....$up to $n$4 termsup to $n$ terms

$Sum=1+2^2-1+2^3-1+2^4-1+.........$ up to $n$ terms

$Sum=1-1-1-1-......upto(n-1)term+2^2+2^3+2^4......$up to $n-1$ terms

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

$Sum=1-n+1+2^2{2}^{n-1}-2^2$

$Sum=2-n+2^{n+1}-4$

$Sum=2^{n+1}-n-2$

Hence, this is the answer.