What is the sum of the first 20 terms of the | vedclass

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(C) The given sequence is $0.7, 0.77, 0.777, \dots$ up to $n=20$ terms.We can write the $n$-th term as $a_n = \frac{7}{9} (1 - 10^{-n})$.The sum of th

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

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(C) The given sequence is $0.7, 0.77, 0.777, \dots$ up to $n=20$ terms.We can write the $n$-th term as $a_n = \frac{7}{9} (1 - 10^{-n})$.The sum of the first $n$ terms is $S_n = \sum_{k=1}^{n} \frac{7}{9} (1 - 10^{-k})$.$S_n = \frac{7}{9} [\sum_{k=1}^{n} 1 - \sum_{k=1}^{n} 10^{-k}]$.$S_n = \frac{7}{9} [n - \frac{10^{-1}(1 - 10^{-n})}{1 - 10^{-1}}]$.$S_n = \frac{7}{9} [n - \frac{1}{10} \cdot \frac{1 - 10^{-n}}{9/10}] = \frac{7}{9} [n - \frac{1}{9} (1 - 10^{-n})]$.For $n=20$,$S_{20} = \frac{7}{9} [20 - \frac{1}{9} (1 - 10^{-20})]$.$S_{20} = \frac{7}{81} [180 - 1 + 10^{-20}] = \frac{7}{81} (179 + 10^{-20})$.

What is the sum of the first $20$ terms of the sequence $0.7, 0.77, 0.777, \dots$?

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