Aman, we can calculate the sum by making the series in form of a geometric progression (GP). The answer to this query is calculated to be 8/81 [10^(n+1)- 9n -10]. The solution of your query is as follows:--

Sum of series is 8+88+888+----+nterms

which can be expressed as;

8[1+11+111+---+n terms]

or, 8/9 [9+99+999+----+n terms]

or, 8/9 [(10-1)+ (10^2-1)+ (10^3-1)+----+n terms]

or, 8/9 [ ( 10^1+ 10^2+ 10^3+----+10^n )-(1*n) ]     (the darkened series is a GP now)

or, 8/9 [ {10*(10^n-1)/9}  -n ]              ( using Sum of a GP formula)

which gives on rearranging,  8/81 [10^(n+1)- 9n -10] as the result.

Hope this helps you.