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. 1
B. 2
C. 3
D. 4

Answer 1

Correct Answer - 2
` a_(n) = a^(n) - beta^(n)`
Also `alpha^(8)` on both sides
`rArr alpha ^(10) - 6alpha ^(9) - 2 alpha^(8) = 0` (1)
Similarly ` beta ^(10) - 6 beta^(9) - 2 beta ^(8) = 0 ` (2)
Subtracting (2) from (1) we have
` alpha ^(10) - beta^(10) - 6 (alpha ^(9) - beta^(9)) = 2 (alpha ^(8) - beta^(8))`
`rArr a_(10) - 6 a_(9) = 2a_(8)`
`rArr (a_(10) - 2 a_(8))/(2a_(9)) = 3 `