If $$\alpha, \beta$$ are the roots of the equation, $$x^2 - x - 1 = 0$$ and $$S_n = 2023\alpha^n + 2024\beta^n$$, then

Since $$\alpha, \beta$$ are roots of $$x^2 - x - 1 = 0$$, we have $$\alpha^2 = \alpha + 1$$ and $$\beta^2 = \beta + 1$$.

In general, $$\alpha^n = \alpha^{n-1} + \alpha^{n-2}$$ and $$\beta^n = \beta^{n-1} + \beta^{n-2}$$.

$$S_n = 2023\alpha^n + 2024\beta^n$$

$$S_{n-1} + S_{n-2} = 2023(\alpha^{n-1} + \alpha^{n-2}) + 2024(\beta^{n-1} + \beta^{n-2})$$

$$= 2023\alpha^n + 2024\beta^n = S_n$$

So $$S_n = S_{n-1} + S_{n-2}$$ for all $$n$$. In particular, $$S_{12} = S_{11} + S_{10}$$.

The answer corresponds to Option (2).