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Let alpha and beta be the roots of x^2 -...

Let `alpha` and `beta` be the roots of `x^2 - x- 1=0` , with `alpha gt beta` . For all positive integers n , define `a_n=(alpha^n-beta^n)/(alpha-beta), n ge1`
`b_(1) = 1 " and " b_(n) = a_(n-1) + a_(n+1), n ge 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 ge 1`

B

`sum_(n=1)^(oo)(a_n)/(10^n)=10/89`

C

`b_n=alpha^n+beta^n` for all `n ge 1`

D

`sum_(n=1)^(oo)b_n/(10^n)=8/89`

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