Question

Find the sum of the following series up to $n$ terms: $5+55+555+....$

Answer 1

$\text{Let}{S}_n=5+55+555+...ton\text{terms}=\frac{5}{9}[9+99+999+...tonterms]=\frac{5}{9}\left[\right(10-1)+({10}^2-1)+({10}^3-1)+..tonterms]=\frac{5}{9}\left[\right(10+{10}^2+{10}^3+...nterms)-(1+1+...nterms\left)\right]=\frac{5}{9}[\frac{10({10}^n-1)}{10-1}-n]=\frac{5}{9}[\frac{10({10}^n-1)}{9}-n]=\frac{50}{81}({10}^n-1)-\frac{5n}{9}$