If α and β are the roots of the equation

Question:

If $\alpha$ and $\beta$ are the roots of the equation $375 x^{2}-25 x-2=0$, then

$\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \alpha^{r}+\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \beta^{r}$ is equal to :

  1. $\frac{21}{346}$

  2. $\frac{29}{358}$

  3. $\frac{1}{12}$

  4. $\frac{7}{116}$

Correct Option: , 3

Solution:

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

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

$\Rightarrow\left(\alpha+\alpha^{2}+\ldots\right.$ upto infinite terms $)+\left(\beta+\beta^{2}\right.$

$+\ldots$ upto infinite terms $)==\frac{\alpha}{1-\alpha}+\frac{\beta}{1-\beta}=\frac{1}{12}$

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