If α and β be the roots of x2−6x−2=0 with α>β, Qn=αn−βn for n≥1, then the value of a10−2a82a9 is

an=αn−βn α2−6α−2=0 α10−6α9−2α8=0___(1) (multiply by α8)β10−6β9−2β8=0____(2) (2)−(1)⇒α10−β10−6(α9−β9)=2(α8−β8) ⇒a10−6a9=2a8⇒a10−2a82a9=3

If α and β be the roots of x2−6x−2=0 with α>β, Qn=αn−βn for n≥1, then the value of a10−2a82a9 is

an=αn−βn α2−6α−2=0 α10−6α9−2α8=0___(1) (multiply by α8)β10−6β9−2β8=0____(2) (2)−(1)⇒α10−β10−6(α9−β9)=2(α8−β8) ⇒a10−6a9=2a8⇒a10−2a82a9=3

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If $α a n d β$ be the roots of $x^{2} - 6 x - 2 = 0$ with $α > β, Q_{n} = α^{n} - β^{n}$ for $n \geq 1,$ then the value of $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ is

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Detailed Solution

$a n = α^{n} - β^{n}$
$α^{2} - 6 α - 2 = 0$
$α^{10} - 6 α^{9} - 2 α^{8} = 0___(1)$
(multiply by $α^{8}$ )
$β^{10} - 6 β^{9} - 2 β^{8} = 0____(2)$
$(2) - (1) \Rightarrow α^{10} - β^{10} - 6 (α^{9} - β^{9}) = 2 (α^{8} - β^{8})$
$\Rightarrow a_{10} - 6 a_{9} = 2 a_{8} \Rightarrow \frac{a_{10} - 2 a_{8}}{2 a_{9}} = 3$