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} 

$=\frac{5}{9}\{9+99+999+\dots \text{ to }\mathrm{n}\text{ terms }\}$

[multiply numerator and denominator by 9]

$\begin{array}{l}=\frac{5}{9}\left\{\right(10-1)+(100-1)+(1000-1)+\dots \text{ to }\mathrm{n}\text{ terms }\}\\ =\frac{5}{9}\{10+100+1000+\dots +\mathrm{n}\text{ terms }-(1+1+\dots +\mathrm{n}\text{ terms }\left)\right\}\\ =\frac{5}{9}\left\{\frac{10\left({10}^{\mathrm{n}}-1\right)}{10-1}-\mathrm{n}\right\}\\ =\frac{5}{81}\left\{{10}^{\mathrm{n}+1}-10-9\mathrm{n}\right\}\end{array}$