α,β are the roots of x2−10x−8=0 with α>β. If an=αn−βn for n∈ℕ , then the value of a10−8a85a9

x2−10x−8=0⇒x2−85x=2α,β are the roots ⇒α2−85α=2⇒α2−8=10α and β2−8=10β ∴ a10−8a85a9=(α10−β10)−8(α8−β8)5(a9−β9) =α8(α2−8)−β8(β2−8)5(α9−β9)=10(α9−β9)5(α9−β9)=2

α,β are the roots of x2−10x−8=0 with α>β. If an=αn−βn for n∈ℕ , then the value of a10−8a85a9

x2−10x−8=0⇒x2−85x=2α,β are the roots ⇒α2−85α=2⇒α2−8=10α and β2−8=10β ∴ a10−8a85a9=(α10−β10)−8(α8−β8)5(a9−β9) =α8(α2−8)−β8(β2−8)5(α9−β9)=10(α9−β9)5(α9−β9)=2

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$α, β$ are the roots of $x^{2} - 10 x - 8 = 0$ with $α > β$ . If $a_{n} = α^{n} - β^{n}$ for $n \in ℕ$ , then the value of $\frac{a_{10} - 8 a_{8}}{5 a_{9}}$

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Detailed Solution

$x^{2} - 10 x - 8 = 0 \Rightarrow \frac{x^{2} - 8}{5 x} = 2$
$α, β$ are the roots $\Rightarrow \frac{α^{2} - 8}{5 α} = 2 \Rightarrow α^{2} - 8 = 10 α$ and $β^{2} - 8 = 10 β$
$∴ \frac{a_{10} - 8 a_{8}}{5 a_{9}} = \frac{(α^{10} - β^{10}) - 8 (α^{8} - β^{8})}{5 (a^{9} - β^{9})} = \frac{α^{8} (α^{2} - 8) - β^{8} (β^{2} - 8)}{5 (α^{9} - β^{9})} = \frac{10 (α^{9} - β^{9})}{5 (α^{9} - β^{9})} = 2$