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

$x^2-x-1=0$
$S_n=2023{\alpha }^n+2024{\beta }^n$
${\mathrm{S}}_{\mathrm{n}-1}+{\mathrm{S}}_{\mathrm{n}-2}=2023{\alpha }^{\mathrm{n}-1}+2024{\beta }^{\mathrm{n}-1}+2023{\alpha }^{\mathrm{n}-2}+2024{\beta }^{\mathrm{n}-2}$
$=2023{\alpha }^{\mathrm{n}-2}[1+\alpha ]+2024{\beta }^{\mathrm{n}-2}[1+\beta ]$
$=2023{\alpha }^{\mathrm{n}-2}\left[{\alpha }^2\right]+2024{\beta }^{\mathrm{n}-2}\left[{\beta }^2\right]$
$=2023{\alpha }^n+2024{\beta }^n$
$S_{n-1}+S_{n-2}=S_n$
$\text{Put }\mathrm{n}=12$
${\mathrm{S}}_{11}+{\mathrm{S}}_{10}={\mathrm{S}}_{12}$