Complex Numbers Question - None

Question

Let i = √(-1), define a sequence of complex numbers by z₁ = 0, z_n+1 = z_n² + i for n ≥ 1. In the complex plane, how far from the origin is z₁₁₁?

Accepted Answer

Step 1: Calculate the first few terms of the sequence:

$z_1 = 0$

$z_2 = z_1^2 + i = 0 + i = i$

$z_3 = z_2^2 + i = i^2 + i = -1 + i$

$z_4 = z_3^2 + i = (-1+i)^2 + i = 1 - 2i - 1 + i = -i$

$z_5 = z_4^2 + i = (-i)^2 + i = -1 + i$

Step 2: Observe the pattern. For n ≥ 3, the sequence becomes periodic: $z_3 = z_5 = -1 + i$, and $z_4$ alternates.

Step 3: Since 111 is odd and 111 ≥ 3, we have $z_{111} = -1 + i$.

Step 4: Calculate the distance from the origin:

$|z_{111}| = |-1 + i| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$

∴ Answer is (b).

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