For given quadratic $375x^2-25x-2=0,$ its roots are,

$\alpha ,\beta =\frac{25\pm \sqrt{{25}^2+2\times 4\times 375}}{2\times 375}\Rightarrow \left|\alpha \right|<1,\left|\beta \right|<1$

Also, $\alpha +\beta =\frac{25}{375},$ $\alpha \beta =\frac{-2}{375}$

Now $\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\sum _{r=1}^n{\alpha }^r+\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\sum _{r=1}^n{\beta }^r$

$=\frac{\alpha }{1-\alpha }+\frac{\beta }{1-\beta }$ $\left[\begin{array}{c}a+ar+a{r}^2+.....\mathrm{\hspace{0.17em}}\mathrm{infinite}\hspace{0.17em}\hspace{0.17em}\mathrm{terms}\\ =\frac{a}{1-r}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{if}\hspace{0.17em}\hspace{0.17em}\left|r\right|<1\end{array}\right]$

$=\frac{\left(\alpha +\beta \right)-2\alpha \beta }{1-\left(\alpha +\beta \right)+\alpha \beta }=\frac{\frac{25}{375}+\frac{4}{375}}{1-\frac{25}{375}-\frac{2}{375}}=\frac{29}{348}=\frac{1}{12}$