JEE MAIN - Mathematics (2024 - 27th January Evening Shift - No. 7)
If $$\alpha, \beta$$ are the roots of the equation, $$x^2-x-1=0$$ and $$S_n=2023 \alpha^n+2024 \beta^n$$, then :
$$2 S_{12}=S_{11}+S_{10}$$
$$S_{12}=S_{11}+S_{10}$$
$$S_{11}=S_{10}+S_{12}$$
$$2 S_{11}=S_{12}+S_{10}$$
Explanation
$$\begin{aligned}
& x^2-x-1=0 \\
& S_n=2023 \alpha^n+2024 \beta^n \\
& S_{n-1}+S_{n-2}=2023 \alpha^{n-1}+2024 \beta^{n-1}+2023 \alpha^{n-2}+2024 \beta^{n-2} \\
& =2023 \alpha^{n-2}[1+\alpha]+2024 \beta^{n-2}[1+\beta] \\
& =2023 \alpha^{n-2}\left[\alpha^2\right]+2024 \beta^{n-2}\left[\beta^2\right] \\
& =2023 \alpha^n+2024 \beta^n \\
& S_{n-1}+S_{n-2}=S_n \\
& P_{u t} n=12 \\
& S_{11}+S_{10}=S_{12}
\end{aligned}$$
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