Let α and β be the roots of x2−8x−2=0. If an=αn−βn for n≥1, then the value of a10−2a84a9 is

α2−6α−2=0, multiply by α8 on both sides⇒α10−8α9−2α8=0⇒β10−8β9−2β8=0∴(α10−β10)−8(α9−β9)−2(α8−β8)=0⇒a10−8a9−2a8=0⇒a10−2a84a9=2

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Let $α a n d β$ be the roots of $x^{2} - 8 x - 2 = 0.$ If $a_{n} = α^{n} - β^{n}$ for $n \geq 1,$ then the value of $\frac{a_{10} - 2 a_{8}}{4 a_{9}}$ is

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answer is 2.

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Detailed Solution

$α^{2} - 6 α - 2 = 0,$ multiply by $α^{8}$ on both sides

$\begin{array}{l} \Rightarrow α^{10} - 8 α^{9} - 2 α^{8} = 0 \\ \Rightarrow β^{10} - 8 β^{9} - 2 β^{8} = 0 \\ ∴ (α^{10} - β^{10}) - 8 (α^{9} - β^{9}) - 2 (α^{8} - β^{8}) = 0 \\ \Rightarrow a_{10} - 8 a_{9} - 2 a_{8} = 0 \\ \Rightarrow \frac{a_{10} - 2 a_{8}}{4 a_{9}} = 2 \end{array}$