If `alpha` is a non-real fourth root of unity, then the value of `alpha^(4n-1)+alpha^(4n-2)+alpha^(4n-3), n in N` is

Options

A. 0
B. `-1`
C. 3
D. none of these

Answer

B

Explanation

We know that fourth roots of unity are `+-1, +-i`. Clearly, non-real fourth roots of unity are `+-i`.
Now, `alpha^(4n-1)+alpha^(4n-2)+alpha^(4n-3)`
`=alpha^(4n(alpha^(-1)+alpha^(-2)+alpha^(-3))`
`=alpha^(-1)+alpha^(-2)+alpha^(-3)` `[therefore alpha^(4n)=1]`
`=alpha^(4)(alpha^(-1)+alpha^(-2)+alpha^(-3))=alpha^(3)+alpha^(2)+alpha` `[therefore alpha^(4)=1]`
`=alpha(alpha^(3)-1)/(alpha-1) = (alpha^(4)-alpha)/(alpha-1)=(1-alpha)/(alpha-1)=-1` `[therefore alpha ne 1]`