若z=根号2/1-i,则z^100+z^50+1的值是

题目

若z=根号2/1-i,则z^100+z^50+1的值是
A.1
B.-1
C.-i
D.i

答案

z=√2/(1-i)=(√2/2)(1+i)
z²=(√2/2)²(1+i)²=(1/2)(1+2i-1)=i
z⁴=(z²)²=i²=-1
z^100+z^50+1
=(z⁴)^25 +[(z⁴)^12]×z²+1
=(-1)^25+[(-1)^12]i +1
=-1+i+1
=i
选D.