If alpha,beta are roots of x^2 -6x-2=0 and alpha>beta , if a_n =(alpha)^n - (beta)^n , n geq 1 then (a_10 - 2a_8)/(2a_9)?

1 Answer
Mar 7, 2018

3

Explanation:

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