The sum of the first 20 terms of the sequence | vedclass

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(C) Let $S_{20} = 0.7 + 0.77 + 0.777 + \dots$ up to $20$ terms. $S_{20} = 7[0.1 + 0.11 + 0.111 + \dots 	ext{ up to } 20 	ext{ terms}]$. Multiply and

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

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(C) Let $S_{20} = 0.7 + 0.77 + 0.777 + \dots$ up to $20$ terms. $S_{20} = 7[0.1 + 0.11 + 0.111 + \dots 	ext{ up to } 20 	ext{ terms}]$. Multiply and divide by $9$: $S_{20} = \frac{7}{9}[0.9 + 0.99 + 0.999 + \dots 	ext{ up to } 20 	ext{ terms}]$. $S_{20} = \frac{7}{9}[(1 - 10^{-1}) + (1 - 10^{-2}) + (1 - 10^{-3}) + \dots + (1 - 10^{-20})]$. $S_{20} = \frac{7}{9}[20 - (10^{-1} + 10^{-2} + \dots + 10^{-20})]$. The sum of the geometric series inside the bracket is $\frac{0.1(1 - 0.1^{20})}{1 - 0.1} = \frac{0.1(1 - 10^{-20})}{0.9} = \frac{1}{9}(1 - 10^{-20})$. $S_{20} = \frac{7}{9}[20 - \frac{1}{9}(1 - 10^{-20})] = \frac{7}{9}[\frac{180 - 1 + 10^{-20}}{9}] = \frac{7}{81}(179 + 10^{-20})$.

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

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