Question

Find the sum to n terms of the series.
$3+5+9+17+33+$__________.

Answer 1

Let $S=3+5+9+17+33+....+T_n$

$S=3+5+9+17+...+T_{n-1}{T}_n$

On subtracting

$0=3+2+4+8+16+.....+T_n-T_{n-1}-T_n$

So, $T_n=3+2+4+8+16+32+....$

$T_n=3+\frac{2(2^{n-1}-1)}{2-1}=1+2^n$

$S_n=\sum T_n=\sum (1+2^n)=\sum \left(1\right)+\sum \left(2^n\right)$

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