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

${\mathrm{S}}_{\mathrm{n}}=5+55+555+\dots \text{ to n terms }$

$\begin{array}{l}=\frac{5}{9}(9+99+999+\dots \text{ to }\mathrm{n}\text{ terms })\\ =\frac{5}{9}\left[\left(10-\underset{ }{\left.\underset{ }{1})+\left({10}^2-1\right)+\left({10}^3-1\right)+\dots +\left({10}^{\mathrm{n}}-1\right)\right]}\right.\right.\\ =\frac{5}{9}\left[\left(10+{10}^2+\dots \text{ to }\mathrm{n}\text{ term }\right)-\mathrm{n}\right] \left(sum of n terms in G.P=\frac{a\left(r^n-1\right)}{r-1}\right)\\ =\frac{5}{9}\left[\frac{10\left({10}^n-1\right)}{9}-n\right]\\ =\frac{5}{81}\left[10\left({10}^{\mathrm{n}}-1\right)-9\mathrm{n}\right]\\ =\frac{5}{81}\left[{10}^{\mathrm{n}+1}-9\mathrm{n}-10\right]\end{array}$