Let ` alpha and beta ` be the roots of ` x^(2) - 6x - 2 = 0 , ` with ` alpha gt beta `. If
` alpha _(n) = a^(n) - beta^(n) ` for `a ne 1` , then the values of `(a_(10) - 2a_(8))/(2a_(9))` is

Options

A. 3
B. -3
C. 6
D. -6

Answer

A

Explanation

It is given that `alpha and beta` are roots of the equation `x^(2)-6x-2=0`.
`therefore" "alpha+beta = 6 and alpha beta = - 2" "...(i)`
We have, `a_(n) = alpha^(n)-beta^(n)`.
`therefore" "(a_(10)-2a_(8))/(2a_(9))=((alpha^(10)-beta^(10))-2(alpha^(8)-beta^(8)))/(2(alpha^(9)-beta^(9)))`
`" "=((alpha^(10)-beta^(10))-2(alpha^(8)-beta^(8)))/(2(alpha^(9)-beta^(9)))" "["usig (i)"]`
`" "=(alpha^(9)(alpha-beta)-beta^(9)(alpha+beta))/(2(alpha^(9)-beta^(9)))`
`" "=((alpha+beta)(alpha^(9)-beta^(9)))/(2(alpha^(9)-beta^(9)))=(alpha+beta)/(2)=(6)/(2)=3`