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, $S_n = 2 + 5 +14 + 41 + .......+ x_n$
$S_n = 2 + 5 +14 + .......+ x_{n-1} + x_n$
$0 = 2 +[3+ 9 + 27 +..........to (n -1) terms] - x_n$
$\therefore x_{n}=2+\frac{3\left(3^{n-1}-1\right)}{3-1}=\frac{1}{2}+\frac{1}{2}.3^{n}$
$\therefore S_{n}=\sum x_{n}=\frac{1}{2} \sum1+\frac{1}{2} \sum 3^{n}$
$=\frac{1}{2}n+\frac{1}{2}\left(3+3^{2}+3^{3}+......+3^{n}\right)$
$=\frac{n}{2}+\frac{1}{2}. \frac{3\left(3^{n}-1\right)}{3-1}=\frac{n}{2}+\frac{3}{4}\left(3^{n}-1\right)$