已知(1+i)2n1−i+(1−i)2n1+i=2n，则最小正整数n=．

Question

已知((1+i)2n)/(1−i)+((1−i)2n)/(1+i)=2ⁿ，则最小正整数n=______．

Answer 1

((1+i)2n)/(1−i)+((1−i)2n)/(1+i)=(2n•in)/(1−i)+(2n•(−i)n)/(1+i)=2ⁿ（(in)/(1−i)+(in)/(1+i)）=2ⁿ（(in)/(1−i)+((−i)n)/(1+i)）
=2ⁿ(in+(−i)n+in+1+(−i)n+1)/((1−i)(1+i))=2ⁿ(in+(−i)n+in+1+(−i)n+1)/(2)=2ⁿ．
最小正整数n=3．
故答案为：3．