Question

The sum of first $20$ terms of the sequence $0.7,0.77,0.777,....,$ is

• A. $\frac{7}{81}(179-10^{-20})$
• B. $\frac{7}{9}(99-10^{-20})$
• C. $\frac{7}{81}(179+10^{-20})$
• D. $\frac{7}{9}(99+10^{-20})$

Answer 1

Answer: (C)

Let $S=0.7+0.77+0.777+\dots$
$=\frac{7}{10}+\frac{77}{10^2}+\frac{777}{10^3}+\dots$ upto $20$ terms
$=7[\frac{1}{10}+\frac{11}{10^2}+\frac{111}{10^3}+\dots$ upto $20$ terms $]$
$=\frac{7}{9}[\frac{9}{10}+\frac{99}{100}+\frac{999}{1000}+\dots$ upto $20$ terms $]$
$=\frac{7}{9}[\left(1-\frac{1}{10}\right)+\left(1-\frac{1}{10^2}\right)+\left(1-\frac{1}{10^3}\right)$
$+\dots +$ upto $20$ terms ]
$=\frac{7}{9}[(1+1+\dots +$ upto $20$ terms $)$
$-(\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+\dots +$ upto $20$ terms $)]$
$=\frac{7}{9}\left[20-\frac{\frac{1}{10}\left\{1-{\left(\frac{1}{10}\right)}^{20}\right\}}{1-\frac{1}{10}}\right]$
$[\because \sum _{i=1}^{20}=20$ and sum of $n$ terms of $]GP,S_n=\frac{a\left(1-r^n\right)}{1-r}$ when $(r<1)$
$=\frac{7}{9}\left[20-\frac{1}{9}\left\{1-{\left(\frac{1}{10}\right)}^{20}\right\}\right]$
$=\frac{7}{9}\left[\frac{179}{9}+\frac{1}{9}{\left(\frac{1}{10}\right)}^{20}\right]=\frac{7}{81}\left[179+(10{)}^{-20}\right]$