Let `alpha` and `beta` be the roots of `x^2-6x-2=0` with `alpha>beta` if `a_n=alpha^n-beta^n` for ` n>=1` then the value of `(a_10 - 2a_8)/(2a_9)`

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

`:'alpha^(2)-6alpha-2=0impliesalpha^(2)-2=6alpha`…..i
and `beta^(2)-6beta-2=0impliesbeta^(2)-2=6beta`…………ii
`:.(a_(10)-2a_(8))/(2a_(9))=((alpha^(10)-beta^(10))-2(alpha^(8)-beta^(8)))/(2(alpha^(9)-beta^(9)))`
`=(alpha^(8)(alpha^(2)-2)(-beta^(8)(beta^(2)-2))/(2(alpha^(9)-beta^(9)))`
`=(alpha^(8).6alpha-beta^(8).6 beta)/(2(alpha^(9)-beta^(9)))` [from Eqs i and ii ]
`=(6(alpha^(9)-beta^(9)))/(2(alpha^(9)-beta^(9)))=3`