The sum of the series 5 +55 +555 +...to n terms is

5 + 55 + 555 +...to n terms=5 {1+ 1 1 + 1 1 1 +...to n terms} =59{9+99+999+… to n terms }[multiply numerator and denominator by 9]=59{(10−1)+(100−1)+(1000−1)+… to n terms }=59{10+100+1000+…+n terms −(1+1+…+n terms )}=591010n−110−1−n=58110n+1−10−9n