Question

若z²+z+1=0,则z^{2 002}+z^{2 003}+z^{2 005}+z^{2 006}的值是(    )

A.2                                B.-2

C.-+i                   D.-±i

Answer 1

思路解析:由z²+z+1=0,不难联系到立方差公式,从而将z得出.

将z²+z+1=0两边乘以(z-1)得z³-1=0,即z³=1(z≠1).则z⁴=z,

z^{2 002}=(z³)⁶⁶⁷·z=z .于是,原式=z^{2 002}(1+z+z³+z⁴)=z(2+2z)=2(z+z²)=-2.

答案:B