Question

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

Answer 1

Correct Answer - A
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`