$5+55+555+\ldots$

Let $S_{n}=5+55+555+\ldots .$ to $n$ terms

$=\frac{5}{9}[9+99+999+\ldots \ldots \text { to } n \text { terms }]$

$=\frac{5}{9}\left[(10-1)+\left(10^{2}-1\right)+\left(10^{3}-1\right)+\ldots \text { to } n \text { terms }\right]$

$=\frac{5}{9}[\left(10+10^{2}+10^{3}+\text { to } n \text { terms }\right)$

$-(1+1+\ldots \text { to } n \text { terms })]$

$=\frac{5}{9}\left[\frac{10\left(10^{n}-1\right)}{10-1}-n\right]$

$=\frac{5}{9}\left[\frac{10\left(10^{n}-1\right)}{9}-n\right]$

$=\frac{50}{81}\left(10^{n}-1\right)-\frac{5 n}{9}$