If z = (1 + I) / (radical 2) Try to find 1 + Z + Z ^ 2 + The value of.. + Z ^ 20

It is easy to get that the modulus of Z is 1, and its direction is the bisector of the first quadrant angle in the complex plane. Then, we can know that the modulus of the power of Z is 1, and there is Z ^ n = - Z ^ (n + 4). Therefore, the sum of the following 16 terms is 0, and after eliminating the terms, we get 1 + Z + Z ^ 2 + .. + Z ^ 20 = 1 + Z + Z ^ 2 + Z ^ 3 + Z ^ 4 = Z + Z ^ 2 + Z ^ 3 = (1 + I) / (radical 2) + I + (- 1 + I)

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