binbinf117
幼苗
共回答了13个问题采纳率:92.3% 举报
证明:因为(-1/2+√3i/2)^3=1,(-1/2-√3i/2)^3=1
(-1/2+√3i/2)^2=-1/2-√3i/2,(-1/2-√3i/2)^2=-1/2+√3i/2
所以(-1/2+√3i/2)^3N=1,(-1/2-√3i/2)^3N=1 (N=1、2、3.)
当n=3N+1时(N=0、1、2、3.)
(-1/2+√3i/2)^n+(-1/2-√3i/2)^n
=(-1/2+√3i/2)^(3N+1)+(-1/2-√3i/2)^(3N+1)
=(-1/2+√3i/2)+(-1/2-√3i/2)
=-1
当n=3N+2时(N=0、1、2、3.)
(-1/2+√3i/2)^n+(-1/2-√3i/2)^n
=(-1/2+√3i/2)^(3N+2)+(-1/2-√3i/2)^(3N+2)
=(-1/2+√3i/2)^2+(-1/2-√3i/2)^2
=-1/2-√3i/2-1/2+√3i/2
=-1
所以原式成立
1年前
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