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Q.
Let $\alpha$ and $\beta$ be the roots of $x^{2}-6 x-2=0$. If
$a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1,$ then the value of $\frac{a_{10}-2 a_{8}}{3 a_{9}}$ is :
JEE MainJEE Main 2021Complex Numbers and Quadratic Equations
Solution:
$ \alpha^{2}-6 \alpha-2=0$
$\alpha^{10}-6 \alpha^{9}-2 \alpha^{8}=0$
Similarly $\frac{\beta^{10}-6 \beta^{9}-2 \beta^{8}=0}{\left(\alpha^{10}-\beta^{10}\right)-6\left(\alpha^{9}-\beta^{9}\right)-2\left(\alpha^{8}-\beta^{8}\right)}=0$
$\Rightarrow a _{10}-6 a _{9}-2 a _{8}=0$
$\Rightarrow \frac{ a _{10}-2 a _{8}}{3 a _{9}}=2$