Question

Sum of the series $3+5+9+17+33+...$ to $n$ terms is

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

Answer 1

Answer: (B)

Given terms are $3,5,9,17,33,....$
Series formed from the differences of the given series is $2,4,8,16,...\in G.P$.
$\therefore ...t_n=A{2}^{n-1}+B$
$\Rightarrow t_1=3,t_2=5$ (Using $3=A+B$ and $5=2A+B$
$\Rightarrow A=2,B=1$)
$\therefore t_n=2^n+1$
$\therefore S_n=\Sigma t_n=\Sigma \left(2^n+1\right)$
$=\Sigma 2^n+\Sigma 1=2\left(2^n-1\right)+n=2^{n+1}+n-2$
Short Cut Method : $S_n=3+5+9+17+33+.....$
$=\left(2+1\right)+\left(2^2+1\right)+\left(2^3+1\right)+\left(2^4+1\right)+......$
$=\left(2+2^2+2^3+2^4+...nterms\right)+n$
$=2\left(2^n-1\right)+n=2^{n+1}+n-2$