Sequence And Series Question 362
Question: The sum to n terms of the series 2+5+14+41+……..is
Options:
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
Show Answer
Answer:
Correct Answer: D
Solution:
[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({3^{n-1}}-1)}{3-1}=\frac{1}{2}+\frac{1}{2}\cdot{3}^{n} $
$ \therefore S_{n}=\sum{x_{n}=\frac{1}{2}\sum{1}+\frac{1}{2}\sum{3^{n}}}} $ $ =\frac{n}{2}+\frac{1}{2}.\frac{3(3^{n}-1)}{3-1}=\frac{n}{2}+\frac{3}{4}(3^{n}-1) $
BETA
