Let α and β be the roots of the equation, 5 x 2 + 6 x - 2 = 0 . If S n = α n + β n , n = 1 , 2 , 3 , . . . . , then

Question

Let α and β be the roots of the equation, 5 x 2 + 6 x - 2 = 0 . If S n = α n + β n , n = 1 , 2 , 3 , . . . . , then, 6 S 6 + 5 S 5 = 2 S 4, 5 S 6 + 6 S 5 + 2 S 4 = 0, 5 S 6 + 6 S 5 = 2 S 4, 6 S 6 + 5 S 5 + 2 S 4 = 0

MARKS App · Accepted Answer

If α , β are roots of quadratic equation 5 x 2 + 6 x - 2 = 0 ⇒ 5 α 2 + 6 α - 2 = 0 ⇒ 5 α 2 + 6 α = 2 ⇒ 5 β 2 + 6 β - 2 = 0 ⇒ 5 β 2 + 6 β = 2 5 S 6 + 6 S 5 = 5 α 6 + β 6 + 6 α 5 + β 5 = α 4 5 α 2 + 6 α + β 4 5 β 2 + 6 β = 2 α 4 + β 4 = 2 S 4

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