Let α and β be the roots of x² - 6x - 2 = 0, with α > β. If aₙ = αⁿ - βⁿ for n ≥ 1, then the value of (a₁₀ - 2a₈) / 2a₉ is?

Since α and β are the roots (solutions) of the equation x² - 6x - 2 = 0
So, it will satisfy the equation
α² - 6α - 2 = 0 (1)
β² - 6β - 2 = 0 (2)
Now, we have to find the value of (a₁₀ - 2a₈) / 2a₉
Given, aₙ = αⁿ - βⁿ
So, a₁₀ = α¹⁰ - β¹⁰ . (3)
So, to get the values of α¹⁰ and β¹⁰, multiplying equation (1) and (2) by α⁸ and β⁸ respectively.
α¹⁰ - 6α⁹ - 2α⁸ = 0
or , α¹⁰ = 6α⁹ + 2α⁸ . (4)
β¹⁰ - 6β⁹ - 2β⁸ = 0
or, β¹⁰ = 6β⁹ + 2β⁸ . (5)
So, using equation(4) and (5) in (3), we get
a₁₀ = 6α⁹ + 2α⁸ - (6β⁹ + 2β⁸)
a₁₀ = 6α⁹ + 2α⁸ - 6β⁹ - 2β⁸
= 6(α⁹ - β⁹) + 2(α⁸ - β⁸)
a₁₀ = 6a₉ + 2a₈
a₁₀ - 2a₈ = 6a₉
(a₁₀ - 2a₈) / 2a₉ = 3