Q51 / 155 · WBJEE 2021 · Mathematics

Let $\alpha, \beta$ be the roots of the equation $x^{2}-6 x-2=0$ with $\alpha>\beta$. If $a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1$, then the value of $\frac{a_{10}-2 a_{8}}{2 a_{9}}$ is

  • A
    1
  • B
    2
  • Correct answer
    C
    3
  • D
    4

Solution

$x^{2}-6 x-2=0$ $x^{n}-6 x^{n-1}-2 x^{n-2}=0$ $\Rightarrow x^{n}-2 x^{n-2}=6 x^{n-1}$ for $n=10$ $x^{10}-2 x^{8}=6 x^{9}$ $\alpha^{10}-2 \alpha^{8}=6 \alpha^{9}$ $\beta^{10}-2 \beta^{8}=6 \beta^{9}$ $\left(\alpha^{10}-\beta^{10}\right)-2\left(\alpha^{8}-\beta^{8}\right)=6\left(\alpha^{9}-\beta^{9}\right)$ $\Rightarrow a_{10}-2 a_{8}=6 a_{9}$ $\Rightarrow \frac{a_{10}-2 a_{8}}{2 a_{9}}=3$

Hence, the correct answer is option: 'C'.

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