Find the sum to $n$ terms
$2 + 22 + 222 + . . . .$

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Solution

$S = 2 + 22 + 222 + . . . n t e r m s S = \frac{2}{9} (9 + 99 + 999 + . . . . n t e r m s) S = \frac{2}{9} ((10 - 1) + (10^{2} - 1) + (10^{3} - 1) + . . . n t e r m s) S = \frac{2}{9} (10 + 10^{2} + . . . . + 10^{n} - n) S = \frac{2}{9} (\frac{10^{n + 1} - 1}{9} - 1 - n) = \frac{2}{9} (\frac{10^{n + 1} - 10}{9} - n) S = \frac{2}{81} (10^{n + 1} - 9 n - 10)$

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