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 :
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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