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

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(C) Let the sum be $S = 0.7 + 0.77 + 0.777 + \dots$ up to $20$ terms.$S = 7[0.1 + 0.11 + 0.111 + \dots]$ up to $20$ terms.Multiply and divide by $9$:$

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

Vedclass · Answer

(C) Let the sum be $S = 0.7 + 0.77 + 0.777 + \dots$ up to $20$ terms.$S = 7[0.1 + 0.11 + 0.111 + \dots]$ up to $20$ terms.Multiply and divide by $9$:$S = \frac{7}{9}[0.9 + 0.99 + 0.999 + \dots]$ up to $20$ terms.$S = \frac{7}{9}[(1 - 0.1) + (1 - 0.01) + (1 - 0.001) + \dots]$ up to $20$ terms.$S = \frac{7}{9}[20 - (0.1 + 0.01 + 0.001 + \dots)]$ up to $20$ terms.The term in the bracket is a Geometric Progression with $a = 0.1$,$r = 0.1$,and $n = 20$.Sum $= \frac{a(1 - r^n)}{1 - r} = \frac{0.1(1 - (0.1)^{20})}{1 - 0.1} = \frac{0.1(1 - 10^{-20})}{0.9} = \frac{1}{9}(1 - 10^{-20})$.Substituting back:$S = \frac{7}{9}[20 - \frac{1}{9}(1 - 10^{-20})] = \frac{7}{9}[\frac{180 - 1 + 10^{-20}}{9}] = \frac{7}{81}(179 + 10^{-20})$.

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

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