Let \alpha and \beta be the roots of x2 – 6x – 2 = 0, with a > b. If a_{n} = \left(\alpha\right)^{n} - \left(\beta\right)^{n} for n \geq 1, then the value of \frac{a_{10}

Engineering

Mathematics

Quadratic Equation and Nature of Roots

Question

Let $α$ and $β$ be the roots of x² – 6x – 2 = 0, with a > b. If $a_{n} = α^{n} - β^{n} for n \geq 1$ , then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is :

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$∴$ x² – 6x – 2 = 0 has roots $α, β$

So, a₂ – 2 = 6 $α and β^{2}$ – 2 = 6 $β$

$a_{n} = α^{n} - β^{n}$

So, $\frac{a_{10} - 2 a_{8}}{2 a_{9}} = \frac{(α^{10} - β^{10}) - 2 (α^{8} - β^{8})}{2 (α^{9} - β^{9})} = \frac{α^{8} (α^{2} - 2) - β^{8} (β^{2} - 2)}{2 (α^{9} - β^{9})} = \frac{α^{8} (6 α) - β^{8} (6 β)}{2 (α^{9} - β^{9})} = 3 .$