Let $α, β, w h e r e α < β,$ be the roots of the quadratic equation $x^{2} - 4 x + 2 = 0$ . For each $n \in N,$ define $S_{n} = α^{n} + β^{n - 1} + α^{n - 2} + β^{n - 3}$ . Then which of the following is correct ?

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$\frac{S_{2024} - 3 S_{2023}}{S_{2022} - S_{2021}} = 3$

$\frac{S_{2024} + S_{2021}}{S_{2023}} = 7$

$\frac{S_{2024} - 6 S_{2022}}{S_{2023} - 2 S_{2021}} = 3$

$\frac{S_{2024} + 8 S_{2021}}{S_{2022}} = 14$

answer is B.

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Detailed Solution

$S_{n} + 2 - 4 S_{n} + 1 + 2 S_{n} = 0 \forall n \in N$ so, $S_{2024} = 4 S_{2023} - 2 S_{2022}$ and $S_{2023} = 4 S_{2022} - 2 S_{2021}$

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