Let and be the roots of the equation, 5 x 2 + 6 x - 2 = 0 . If S n = n + 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 $S_{n} = α^{n} + β^{n},$ $n = 1, 2, 3, \dots,$ then

Smart Achievers · Accepted Answer

(3)

$Given α, β are the roots of the equation$ $5 x^{2} + 6 x - 2 = 0 .$

$\Rightarrow α + β = - \frac{6}{5} and α β = - \frac{2}{5}$

$Now, S_{n} = α^{n} + β^{n} \Rightarrow S_{n + 1} = α^{n + 1} + β^{n + 1}$

$S_{n + 1} = (α^{n} + β^{n}) (α + β) - α β (α^{n - 1} + β^{n - 1})$

$= S_{n} (- \frac{6}{5}) - (- \frac{2}{5}) S_{n - 1}$ $[∵ α + β = - \frac{6}{5}, α β = - \frac{2}{5} and α^{n} + β^{n} = S_{n}]$

$\Rightarrow S_{n + 1} = - \frac{6}{5} S_{n} + \frac{2}{5} S_{n - 1}$ $\Rightarrow S_{n + 1} = \frac{2}{5} (S_{n - 1} - 3 S_{n})$

$\Rightarrow 5 S_{n + 1} = 2 (S_{n - 1} - 3 S_{n}),$ $which satisfies option (3).$

Want Us to Email you About Special Offers & Updates?