z1=1+3i,z2=3+i,z1,z,z2的倒数成等差数列,求z

题目

z1=1+3i,z2=3+i,z1,z,z2的倒数成等差数列,求z

答案

z1,z,z2的倒数成等差数列
2/z
=1/z1+1/z3
=1/(1+3i)+1/(3+i)
=(1-3i)/(1+9)+(3-i)/(9+1)
=(2-2i)/5
z=10/(2-2i)
=5/(1+i)
=5(1-i)/(1+1)
所以z=(5-5i)/2