If α and β are the roots of the equation 375 x ^ 2 - 25x - 2 = 0,...

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

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

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

Now $\lim_{n \to \infty}\mathop{∑}_{r = 1}^{n}{\alpha}^{r} + \lim_{n \to \infty}\mathop{∑}_{r = 1}^{n}{\beta}^{r}$

= \dfrac{\alpha}{1 - \alpha} + \dfrac{\beta}{1 - \beta} \left[$\begin{array}{ll}a + ar + a{r}^{2} + . . . . . infiniteterms \\ = \dfrac{a}{1 - r}if \left|r\right| < 1\end{array}$\right]

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