If $α, β$ are the roots of the equation $375 x^{2} - 25 x - 2 = 0$ and $S_{n} = α^{n} + β^{n}$ , then $\underset{n \to \infty}{L t} \sum_{r = 1}^{n} S_{r}$ is

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$\frac{35}{12}$

$\frac{7}{12}$

$\frac{1}{12}$

None of these

answer is B.

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Detailed Solution

$\begin{array}{l} \sum_{r = 1}^{n} S_{r} = (α + β) + (α^{2} + β^{2}) + \dots + (α^{n} + β^{n}) \\ = (α + α^{2} + \dots α^{n}) + (β + β^{2} + \dots β^{n}) \\ \underset{n \to \infty}{L t} \sum_{r = 1}^{n} S_{r} = (α + α^{2} + \dots \infty) + (β + β^{2} + \dots, \infty) \end{array}$

(G.P series) $S_{\infty} = \frac{a}{1 - r}$

$\begin{array}{l} = \frac{α}{1 - α} + \frac{β}{1 - β} \\ = \frac{α - α β + β - α β}{1 - (α + β) + α β} \\ = \frac{α + β - 2 α β}{1 - (α + β) + α β} \\ = \frac{\frac{25}{375} + \frac{4}{375}}{1 - \frac{25}{375} - \frac{2}{375}} \\ = \frac{29}{348} = \frac{1}{12} \end{array}$