Matrices and Determinants 1 Question 15
15. Let $b=6$, with $a$ and $c$ satisfying Eq. (i). If $\alpha$ and $\beta$ are the roots of the quadratic equation $a x^{2}+b x+c=0$, then $\sum_{n=0}^{\infty} (\frac{1}{\alpha}+\frac{1}{\beta})^{n}$ is equal to
(a) 6
(b) 7
(c) $\frac{6}{7}$
(d) $\infty$
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Answer:
Correct Answer: 15. (b)
Solution:
- If $b=6 \Rightarrow a=1$ and $c=-7$
$\therefore a x^{2}+b x+c=0 \Rightarrow x^{2}+6 x-7=0$
$\Rightarrow(x+7)(x-1)=0$
$\therefore \quad x=1,-7$
$ \begin{aligned} \Rightarrow \sum_{n=0}^{\infty} (\frac{1}{1}{-\frac{1}{7})^{n}} & =1+\frac{6}{7}+\frac{6}{7}^{2}+\cdots+\infty=\frac{1}{1-\frac{6}{7}} \ & =\frac{1}{1 / 7}=7 \end{aligned} $
