alpha and beta are roots of x^2-6x-2
So, =>alpha^2-6alpha-2=0
=>alpha^2=6alpha+2
Multiply alpha^8 both sides.
=>alpha^2×color (blue)(alpha^8)=color (blue)(alpha^8) (6alpha+2)
=>alpha^10=6alpha^9+2alpha^8" equation"1
Similarly
=>beta^2=6beta+2
=>beta^10=6beta^9+2beta^8" " "equation"2
We have
=>a_n=alpha^n-beta^n" "("where " alpha>beta)
we have to find
=>(a_10-2a_8)/a_9
Or
=>(alpha^10-beta^10-2(alpha^8-beta^8))/(2a_9)
From equation 1" and "2
=>(6alpha^9+cancel (2alpha^8)-6beta^9-cancel (2beta^8)-cancel (2alpha^8)+cancel (2beta^8))/(2a_9)
=>(6 (alpha^9-beta^9))/(2a_9)
=>(6a_9)/(2a_9)
=>3