Question

Sum of the series $\frac{2}{3}+\frac{8}{9}+\frac{26}{27}+\frac{80}{81}+....$ to $n$ terms is

• 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. $n+\frac{1}{2}(3^{-n}-1)$

Answer 1

Answer: (D)

$T_{n} =\frac{3^{n}-1}{3^{n}}=1-(\frac{1}{3})^{n}$
$S_{n} =n- \sum\limits _{n=1}^{n}(\frac{1}{3})^{n}$
$=n-\frac{\frac{1}{3}[1-(\frac{1}{3})^{n}]}{(1-\frac{1}{3})}$
$=n-\frac{1}{2}(1-3^{-n}) =n+\frac{1}{2} (3^{-n}-1)$