Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$, with $\alpha>\beta$. For all positive integers $n$, define
$$
\begin{aligned}
& a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1 \\
& b_1=1 \text { and } b_n=a_{n-1}+a_{n+1}, n \geq 2
\end{aligned}
$$
Then which of the following options is/are correct?
Select ALL correct options:
A
$\mathrm{a}_1+\mathrm{a}_2+\mathrm{a}_3+\ldots . .+\mathrm{a}_{\mathrm{n}}=\mathrm{a}_{\mathrm{n}+2}-1$ for all $\mathrm{n} \geq 1$
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