由于

111…1

n个1

=

1

9

×

999…9

n个

=

10n-1

9

                       
∴1+11+111+…+

111…1

n个1

=

1

9

[(10-1)+(102-1)+…+(10n-1)]
=

1

9

(10+102+…+10n)-

n

9

=

1

9

•

10(1-10n)

1-10

-

n

9

=

10n+1-9n-10

81