Let α and β are the roots of x2−x−1=0, with α>β. For all positive integers n, define an=αn−βnα−β, n≥1,b1=1 and bn=an−1+an+1,n≥2. Then which of the following options is/are correct?

b n = α n + β n f o r a l l n ≥ 1 , ∑ n = 1 ∞ a n 10 n = 8 89 , ∑ n = 1 ∞ b n 10 n = 12 89

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Let $α$ and $β$ are the roots of $x^{2} - x - 1 = 0,$ with $α > β .$ For all positive integers n, define $a_{n} = \frac{α^{n} - β^{n}}{α - β}$ , $n \geq 1, b_{1} = 1$ and $b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2.$ Then which of the following options is/are correct?

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$b_{n} = α^{n} + β^{n} f o r a l l n \geq 1$

$a_{1} + a_{2} + a_{3} + ....... + a_{n} = a_{n + 2} - 1 f o r a l l n \geq 1$

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

$\sum_{n = 1}^{\infty} \frac{b_{n}}{10^{n}} = \frac{12}{89}$

answer is A, C, D.

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