If $a, \beta$ are roots of the equation $375x^2-25x-2=0$ and $S_n={\alpha }^n+{\beta }^n,$ then $\underset{n\in \mathrm{\infty }}{lim} \sum _{r=1}^n S_r$ is equal to 

It is given that $a, \beta$ are the roots of the equation

$375x^2-25x-2=0$

$\therefore \alpha +\beta =\frac{1}{15}$ and $\alpha \beta =\frac{-2}{375}$

$\begin{array}{l}\therefore \underset{n\to \mathrm{\infty }}{lim} \sum _{r=1}^n S_r=\underset{n\to \mathrm{\infty }}{lim} \sum _{r=1}^n \left({\alpha }^r+{\beta }^r\right)\\ \Rightarrow \underset{n\to \mathrm{\infty }}{lim} \sum _{r=1}^n S_r=\left(\alpha +{\alpha }^2+{\alpha }^3+\dots \mathrm{\infty }\right)+\left(\beta +{\beta }^2+{\beta }^3+\dots +\mathrm{\infty }\right)\\ \Rightarrow \underset{n\to \mathrm{\infty }}{lim} \sum _{r=1}^n S_{+}=\frac{\alpha }{1-\alpha }+\frac{\beta }{1-\beta }[\because |\alpha |<1,|\beta |<1]\\ \Rightarrow \underset{n\to \mathrm{\infty }}{lim} \sum _{r=1}^n S=\frac{\alpha +\beta -2\alpha \beta }{1-(\alpha +\beta )+\alpha \beta }=\frac{29}{348}\end{array}$