Question

The sum to n terms of the series $2+5+14+41+........$ is

• A. $3^{n-1}+8n-3$
• B. $8.3^n+4n-8$
• C. $3^{n+1}+\frac{8}{3}n+1$
• D. None of these

Answer 1

Answer: (D)

Let, $S_n=2+5+14+41+.......+x_n$
$S_n=2+5+14+.......+x_{n-1}+x_n$
$0=2+[3+9+27+..........to(n-1)terms]-x_n$
$\therefore x_n=2+\frac{3\left(3^{n-1}-1\right)}{3-1}=\frac{1}{2}+\frac{1}{2}.3^n$
$\therefore S_n=\sum x_n=\frac{1}{2}\sum 1+\frac{1}{2}\sum 3^n$
$=\frac{1}{2}n+\frac{1}{2}\left(3+3^2+3^3+......+3^n\right)$
$=\frac{n}{2}+\frac{1}{2}.\frac{3\left(3^n-1\right)}{3-1}=\frac{n}{2}+\frac{3}{4}\left(3^n-1\right)$