利用通项求和，求1+11+111+…+111…1n个1之和．

Question

利用通项求和，求1+11+111+…+

111…1

n个1

之和．

Answer 1

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)