Let $$\alpha and \beta$$ be the roots of $$x^2 — x — 1 = 0,$$ with \alpha > \beta$$. For all positive integers n, define
$$a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta},$$ $$n \geq 1,$$
$$b_1 = 1$$ and $$b_n = a_{n - 1} + a_{n + 1},$$ $$n \geq 2.$$
Then which of the following options is/are correct?

A

$$a_1 + a_2 + a_3 + ... + a_n = a_{n + 2} - 1$$ for all $$n \geq 1$$

B

$$\sum_{n = 1}^\infty \frac{a_n}{10^n} = \frac{10}{89}$$

C

$$b_n = \alpha^n + \beta^n$$ for all $$n \geq 1$$

D

$$\sum_{n = 1}^\infty \frac{b_n}{10^n} = \frac{8}{89}$$