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$7 + 77 + 777 + ...... n t e r m s =$

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\frac{70 (10^{n} - 1)}{81} - \frac{7 n}{9}

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

7 (10^{n} - 1) - \frac{7 n}{9}

\frac{7 (10^{n} + 1)}{81} + \frac{7 n}{9}

answer is A.

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Detailed Solution

7 + 77 + 777 + ........ + n t e r m s

$= 7 (1 + 11 + 111 + ......... + n t e r m s)$

$= \frac{7}{9} [9 + 99 + 999 + ........ + n t e r m s]$

$= \frac{7}{9} [(10 - 1) + (100 - 1) + (1000 - 1) + ........... + n t e r m s]$

$= \frac{7}{9} [(10 + 100 + 1000 + ......... + n t e r m s) - (1 + 1 + ......... + n t e r m s)]$

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

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

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

$= \frac{70 (10^{n} - 1)}{81} - \frac{7 n}{9}$

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