What's the fastest approach?

Algebra Level 4

Let α\alpha and β\beta be the roots of x26x2=0x^2 - 6x - 2 = 0, with α>β\alpha > \beta . If an=αnβna_n = \alpha ^n - \beta ^n for n1n\ge 1, then find the value of:

a102a82a9\frac {a_{10} - 2a_8} {2a_9}

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