Question

If $3+5+9+17+33+\dots$ to $n$ terms $=2^{n+1}+n-2$, then $n^{th}$ term of LHS is

• A. $3^n-1$
• B. $2n+1$
• C. $2^n+1$
• D. $3n-1$

Answer 1

The correct option is C $2^n+1$

Let $n^{th}$ term of the given series be $T_n$
$\because$ $3+5+9+17+33+\cdots +T_n=2^{n+1}+n-2$ $\Rightarrow (2^1+1)+(2^2+1)+(2^3+1)+\cdots +(2^n+1)=2^{n+1}+n-2$
$\Rightarrow (2^1+1)+(2^2+1)+(2^3+1)+\cdots +(2^n+1)=(2^{n+1}-2)+n$
$\Rightarrow (2^1+1)+(2^2+1)+(2^3+1)+\cdots +(2^n+1)=\frac{2(2^n-1)}{2-1}+n$
Hence on comparison,
$\Rightarrow T_n=2^n+1$