Question

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

• A. $\frac{21}{346}$
• B. $\frac{29}{358}$
• C. $\frac{1}{12}$
• D. $\frac{7}{116}$

Answer 1

Answer: (C)

$375x^{2} -25x-2=0$
$ \alpha + \beta = \frac{25}{375}, \alpha\beta = \frac{-2}{375} $
$ \Rightarrow \left(\alpha + \alpha^{2} + ..... \text{upto} \, \text{infinite} \, \text{terms}\right) + \left(\beta +\beta^{2} + .... \text{upto} \, \text{infinite} \, \text{terms}\right) $
$= \frac{\alpha}{1-\alpha} + \frac{\beta}{1-\beta} = \frac{1}{12} $