Sequence And Series Question 147
Question: Sum of the series $ \frac{2}{3}+\frac{8}{9}+\frac{26}{27}+\frac{80}{81}+….. $ to n terms is
[Karnataka CET 2001]
Options:
A) $ n-\frac{1}{2}(3^{n}-1) $
B) $ n+\frac{1}{2}(3^{n}-1) $
C) $ n+\frac{1}{2}(1-{3^{-n}}) $
D) $ n+\frac{1}{2}({3^{-n}}-1) $
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Answer:
Correct Answer: D
Solution:
$ T_{n}=\frac{3^{n}-1}{3^{n}}=1-{{( \frac{1}{3} )}^{n}} $ $ S_{n}=n-\sum\limits_{n=1}^{n}{{{( \frac{1}{3} )}^{n}}} $ $ =n-\frac{\frac{1}{3}[ 1-{{( \frac{1}{3} )}^{n}} ]}{( 1-\frac{1}{3} )} $ $ =n-\frac{1}{2}(1-{3^{-n}})=n+\frac{1}{2}({3^{-n}}-1) $ .
BETA
