Let α,β , where α<β, be the roots of the quadratic equation x2−4x+2=0. For each n∈N, define Sn=αn+βn−1+αn−2+βn−3 . Then which of the following is correct?

Sn+2−4Sn+1+2Sn=0∀n∈N so, S2024=4S2023−2S2022 and S2023=4S2022−2S2021

Let α,β , where α<β, be the roots of the quadratic equation x2−4x+2=0. For each n∈N, define Sn=αn+βn−1+αn−2+βn−3 . Then which of the following is correct?

Sn+2−4Sn+1+2Sn=0∀n∈N so, S2024=4S2023−2S2022 and S2023=4S2022−2S2021

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Let $α, β$ , where $α < β$ , 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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$\frac{S_{2024} - 3 S_{2023}}{S_{2022} - S_{2021}} = 3$

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

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

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

answer is B.

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$S_{n + 2} - 4 S_{n + 1} + 2 S_{n} = 0 \forall n \in N$

so, $S_{2024} = 4 S_{2023} - 2 S_{2022}$ and $S_{2023} = 4 S_{2022} - 2 S_{2021}$

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Let α,β , where α<β, be the roots of the quadratic equation x2−4x+2=0. For each n∈N, define Sn=αn+βn−1+αn−2+βn−3 . Then which of the following is correct?

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Let $α, β$ , where $α < β$ , 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?