Let α and β be the roots of equation x 2 - 6 x - 2 = 0 . If a n = α n - β n , ∀ n ≥ 1 , then the value of a 1 0 - 2 a 8 2 a 9 is equal to

Step-by-step solution

Given, α , β are roots of x 2 - 6 x - 2 = 0 . ⇒ α 2 - 6 α - 2 = 0 and β 2 - 6 β - 2 = 0 ⇒ α 2 - 2 = 6 α and β 2 - 2 = 6 β . . . 1 a 1 0 - 2 a 8 2 a 9 = α 10 - β 10 - 2 α 8 - β 8 2 α 9 - β 9 = α 1 0 - 2 α 8 - β 1 0 - 2 β 8 2 α 9 - β 9 = α 8 α 2 - 2 - β 8 β 2 - 2 2 α 9 - β 9 = α 8 6 α - β 8 6 β 2 α 9 - β 9 [ ∵ From (1)] = 6 α 9 - 6 β 9 2 α 9 - β 9 = 3