The sum of the series 6 + 66 + 666 + dots up | vedclass

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(B) Let $S_n = 6 + 66 + 666 + \dots$ up to $n$ terms.$S_n = 6(1 + 11 + 111 + \dots + n 	ext{ terms})$$S_n = \frac{6}{9}(9 + 99 + 999 + \dots + n 	ex

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$2(10^{n+1} - 9n - 10)/27$

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(B) Let $S_n = 6 + 66 + 666 + \dots$ up to $n$ terms.$S_n = 6(1 + 11 + 111 + \dots + n 	ext{ terms})$$S_n = \frac{6}{9}(9 + 99 + 999 + \dots + n 	ext{ terms})$$S_n = \frac{2}{3}((10 - 1) + (10^2 - 1) + (10^3 - 1) + \dots + (10^n - 1))$$S_n = \frac{2}{3}((10 + 10^2 + 10^3 + \dots + 10^n) - (1 + 1 + 1 + \dots + n 	ext{ times}))$Using the sum of a geometric progression formula $\frac{a(r^n - 1)}{r - 1}$:$S_n = \frac{2}{3} \left( \frac{10(10^n - 1)}{10 - 1} - n ight)$$S_n = \frac{2}{3} \left( \frac{10(10^n - 1)}{9} - n ight)$$S_n = \frac{2}{3} \left( \frac{10^{n+1} - 10 - 9n}{9} ight)$$S_n = \frac{2(10^{n+1} - 9n - 10)}{27}$.

The sum of the series $6 + 66 + 666 + \dots$ up to $n$ terms is

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