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

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$\frac{7}{81} (179 - 10^{- 20})$

$\frac{7}{9} (99 + 10^{- 20})$

$\frac{7}{81} (179 + 10^{- 20})$

$\frac{7}{9} (99 - 10^{- 20})$

answer is C.

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Detailed Solution

Let $S = 0.7 + 0.77 + 0.777 + . . . +$ up to 20 terms.

Then,

$\begin{array}{l} S = \frac{7}{10} + \frac{77}{100} + \frac{777}{1000} + \dots . upto 20 terms \\ \Rightarrow S = 7 \{\frac{1}{10} + \frac{11}{100} + \frac{111}{1000} + \dots upto 20 terms\} \\ \Rightarrow S = \frac{7}{9} \{\frac{9}{10} + \frac{99}{100} + \frac{999}{1000} + \dots upto 20 terms\} \end{array}$

$\begin{array}{l} \Rightarrow S = \frac{7}{9} \{(1 - \frac{1}{10}) + (1 - \frac{1}{100}) + (1 - \frac{1}{1000}) + \dots + (1 - \frac{1}{10^{20}})\} \\ \Rightarrow S = \frac{7}{9} \{20 - (\frac{1}{10} + \frac{1}{10^{2}} + \dots + \frac{1}{10^{20}})\} \\ \Rightarrow S = \frac{7}{9} \{20 - \frac{1}{10} (\frac{1 - \frac{1}{10^{20}}}{1 - \frac{1}{10}})\} \\ \Rightarrow S = \frac{7}{9} \{20 - \frac{1}{9} (1 - \frac{1}{10^{20}})\} \\ \Rightarrow S = \frac{7}{9} \{\frac{179}{9} + \frac{1}{9} {(\frac{1}{10})}^{20}\} = \frac{7}{81} (179 + 10^{- 20}) \end{array}$