Let `S_n` denotes the sum of the terms of n series `(1lt=nlt=9)` `1+22+333+.....999999999`, is

Options

A. `S_(n)-S_(n-1)=(1)/(9)(10^(n)-n^(2)+n)`
B. `S_(n)=(1)/(9)(10^(n)-n^(2)+2n-2)`
C. `9(S_(n)-S_(n-1))=n(10^(n)-1)`
D. None of these

Answer

C

Explanation

`:. S_(n)=1+22+333+"..."+ubrace(" nnn""...."n)_(" n terms ")`
`:. S_(n)-S_(n-1)=ubrace(" nnn""...."n)_(" n times ")=ubrace(111"...."1)_(" n times ")`
`=n(10^(n-1)+10^(n-2)+"….."+10+1)=(n(10^(n)-1))/(10-1)`
`:. 9(S_(n)-S_(n-1))=n(10^(n)-1)`