Let α and β 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? [JEE Adv 2019 P1]

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

b_{n}=\alpha^{n}+\beta^{n} for all n \geq 1

\sum_{n=1}^{\infty} {b_{n}}/{10^{n}}= \frac{8}{89}

a_{1}+a_{2}+a_{3}+......a_{n}=a_{n+2}-1 for all n \geq 1

Quadratic Equation and Inequalities

Section: Mathematics

Question : 6 of 23

Marks: +1, -0

Let α and β be the roots of x2−x−1=0,with α>β For all positive integers n, define
an=

αn−βn

α−β

,n.≥1
b1=1 and bn=an−1+an+1,n≥2.
Then which of the following options is/are correct?

[JEE Adv 2019 P1]

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