let p, q be integers and let $\alpha, \beta$ be the roots of the equaion, $x^2-x-1=0$, where $\alpha \neq \beta$. For $n= 0,1,2, \ldots \ldots$, Let $\mathrm{a}_{\mathrm{n}}=\mathrm{p} \alpha^{\mathrm{n}}+\mathrm{q} \beta^{\mathrm{n}}$
FACT : If $a$ and $b$ are rational nubers and $a+b \sqrt{5}=0$, then $a=0=b$.
$a_{12}=$