$ 1+11+111+1111+...+1111...1111 $
$ = \frac{9}{9}+\frac{99}{9}+\frac{999}{9}+...+\frac{99999...99}{9} $
$ = \frac{1}{9} ( 10-1+10^2-1+10^3-1+...+10^n-1 ) $
$ = \frac{1}{9} (10+10^2+10^3+...+10^n -n )$
$ = \frac{1}{9} (\frac{10(10^n-1)}{10-1}-n )$
$ =\frac{10^{n+1}-9n-10}{81} $
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