Sum to 25 terms of the series 0.5 + 0.55 + 0.555 + . . . is:

Let $S=0.5+0.55+0.555+\dots +$ upto $25$ terms

$=\frac{5}{9}[0.9+0.99+0.999+\dots +$up $25$ terms $]$

$\begin{array}{r}=\frac{5}{9}\left[(1-0.1)+\left(1-(0.1{)}^2\right)+\left(1-(0.1{)}^3\right)\right.\\ \left.+\dots +\left(1-(0.1{)}^{25}\right)\right]\end{array}$

$\begin{array}{l}=\frac{5}{9}\left[25-\frac{\left(0.1\right)\left(1-(0.1{)}^{25}\right)}{1-0.1}\right]\\ =\frac{5}{81}\left[224-{10}^{-25}\right]\end{array}$