The value of `alpha^(-n)+alpha^(-2n), n in N` and `alpha` is a non-real cube root of unity, is

Options

A. 3, if n is a multiple of 3
B. `-1`, if n is a mulitiple of 3
C. 2, if n is a multiple of 3
D. none of these

Answer

C

Explanation

Let `alpha=omega`. Then,
`alpha^(-n)+alpha^(-2n)=1/omega^(n)+1/omega^(2n) =(1/omega)^(n) +(1/omega^(2))^(n)`
`rArr alpha^(-n) + alpha^(-2n)= alpha^(2n)+alpha^(n)={{:(2,"if n is a multiple of 3"),(-1, "if n is not a multiple of 3"):}}`