Let α and β be the roots of x 2 − 6 x − 2 = 0 with α β . If a n = α n − β n for n ≥ 1 , then the value of 2 a 9 a 10 − 2 a 8 is

Question

Let α and β be the roots of x 2 − 6 x − 2 = 0 with α > β . If a n ​ = α n − β n for n ≥ 1 , then the value of 2 a 9 ​ a 10 ​ − 2 a 8 ​ ​ is

TestHub · Accepted Answer

2 a 9 ​ a 10 ​ − 2 a 8 ​ ​ = 2 ( α 9 − β 9 ) ( α 10 − β 10 ) − 2 ( α 8 − β 8 ) ​ = 2 ( α 9 − β 9 ) α 8 ( α 2 − 2 ) − β 8 ( β 2 − 2 ) ​ [ ∵ α is root of x 2 − 6 x − 2 = 0 α 2 − 2 = 6 α ] Also, β is root of x 2 − 6 x − 2 = 0 ​ ⇒ β 2 − 2 = 6 β = 2 ( α 9 − β 9 ) α 8 ( 6 α ) − β 8 ( 6 β ) ​ = 2 ( α 9 − β 9 ) 6 ( α 9 − β 9 ) ​ = 3 ​

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