If alpha and beta are the roots of the equati | vedclass

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(A) Given the quadratic equation $375x^2 - 25x - 2 = 0$. By Vieta's formulas,the sum of the roots is $\alpha + \beta = -(\frac{-25}{375}) = \frac{25}{

Vedclass · Accepted Answer

$\frac{1}{12}$

Vedclass · Answer

(A) Given the quadratic equation $375x^2 - 25x - 2 = 0$. By Vieta's formulas,the sum of the roots is $\alpha + \beta = -(\frac{-25}{375}) = \frac{25}{375} = \frac{1}{15}$ and the product of the roots is $\alpha \beta = \frac{-2}{375}$. The expression is $\sum_{r=1}^{\infty} \alpha^r + \sum_{r=1}^{\infty} \beta^r = \frac{\alpha}{1-\alpha} + \frac{\beta}{1-\beta}$. This simplifies to $\frac{\alpha(1-\beta) + \beta(1-\alpha)}{(1-\alpha)(1-\beta)} = \frac{\alpha - \alpha\beta + \beta - \alpha\beta}{1 - (\alpha+\beta) + \alpha\beta}$. Substituting the values: $\frac{(\alpha+\beta) - 2\alpha\beta}{1 - (\alpha+\beta) + \alpha\beta} = \frac{\frac{1}{15} - 2(\frac{-2}{375})}{1 - \frac{1}{15} + (\frac{-2}{375})} = \frac{\frac{25}{375} + \frac{4}{375}}{\frac{375 - 25 - 2}{375}} = \frac{29}{348} = \frac{1}{12}$.

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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