Let α and β be the roots of the equation, 5x^2 + 6x – 2 = 0. If Sn = αn + βn, n = 1, 2, 3,....., then :

Question

Let $\alpha$ and $\beta$ be the roots of the equation, 5x² + 6x – 2 = 0. If S_n = $\alpha$ⁿ + $\beta$ⁿ, n = 1, 2, 3,....., then :

(1) 6S₆ + 5S₅ = 2S₄

(2) 5S₆ + 6S₅ = 2S₄

(3) 5S₆ + 6S₅ + 2S₄ = 0

(4) 6S₆ + 5S₅ + 2S₄ = 0

Answer 1

Ans. (2) 5S₆ + 6S₅ = 2S₄

Sol. 5$\alpha$² + 6$\alpha$ = 2

5S₆ + 6S₅ = $\alpha$⁴(5$\alpha$² + 6$\alpha$) + $\beta$⁴(5$\beta$² + 6$\beta$)

= 2($\alpha$⁴ + $\beta$⁴) = 2S₄