Find the sum of the sequence 7, 77, 777, 7777 | vedclass

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(A) Let the sum be $S_n = 7 + 77 + 777 + 7777 + \ldots$ to $n$ terms. We can write this as: $S_n = \frac{7}{9} [9 + 99 + 999 + 9999 + \ldots 	ext{ to

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$\frac{7}{81}[10(10^n - 1) - 9n]$

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(A) Let the sum be $S_n = 7 + 77 + 777 + 7777 + \ldots$ to $n$ terms. We can write this as: $S_n = \frac{7}{9} [9 + 99 + 999 + 9999 + \ldots 	ext{ to } n 	ext{ terms}]$ $S_n = \frac{7}{9} [(10 - 1) + (10^2 - 1) + (10^3 - 1) + \ldots + (10^n - 1)]$ $S_n = \frac{7}{9} [(10 + 10^2 + 10^3 + \ldots + 10^n) - (1 + 1 + 1 + \ldots + 1 	ext{ to } n 	ext{ terms})]$ The first part is a geometric progression with first term $a = 10$ and common ratio $r = 10$. Using the sum formula $S = \frac{a(r^n - 1)}{r - 1}$: $S_n = \frac{7}{9} [\frac{10(10^n - 1)}{10 - 1} - n]$ $S_n = \frac{7}{9} [\frac{10(10^n - 1)}{9} - n]$ $S_n = \frac{7}{81} [10(10^n - 1) - 9n]$

Find the sum of the sequence $7, 77, 777, 7777, \ldots$ to $n$ terms.

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