Journey with alpha and beta

Algebra Level 4

Let α\alpha and β\beta be the roots of x26x2x^{2} - 6x -2 , with α>β\alpha > \beta. If an=αnβna_{n} = \alpha^{n} - \beta^{n} for n>1n>1, then find the value of a102a82a9\dfrac{a_{10} - 2a_{8}}{2a_{9}}.

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