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(A) Given the quadratic equation $375x^2 - 25x - 2 = 0$.By the properties of roots,the sum of roots $\alpha + \beta = -(-25)/375 = 25/375 = 1/15$ and

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$\frac{1}{12}$

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(A) Given the quadratic equation $375x^2 - 25x - 2 = 0$.By the properties of roots,the sum of roots $\alpha + \beta = -(-25)/375 = 25/375 = 1/15$ and the product of roots $\alpha \beta = -2/375$.We need to evaluate the sum of two infinite geometric series: $S = \sum_{r=1}^{\infty} \alpha^r + \sum_{r=1}^{\infty} \beta^r$.Since $|\alpha| Thus,$S = \frac{\alpha}{1-\alpha} + \frac{\beta}{1-\beta}$.$S = \frac{\alpha(1-\beta) + \beta(1-\alpha)}{(1-\alpha)(1-\beta)} = \frac{\alpha - \alpha\beta + \beta - \alpha\beta}{1 - (\alpha+\beta) + \alpha\beta}$.Substituting the values: $S = \frac{(\alpha+\beta) - 2\alpha\beta}{1 - (\alpha+\beta) + \alpha\beta}$.$S = \frac{1/15 - 2(-2/375)}{1 - 1/15 - 2/375} = \frac{25/375 + 4/375}{(375 - 25 - 2)/375} = \frac{29/375}{348/375} = \frac{29}{348} = \frac{1}{12}$.

If $\alpha$ and $\beta$ are the roots of the equation $375x^2 - 25x - 2 = 0$,then $\lim_{n \to \infty} \sum_{r=1}^n \alpha^r + \lim_{n \to \infty} \sum_{r=1}^n \beta^r$ is equal to

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