First slide

Algebra of complex numbers

Question

Let $i = \sqrt{- 1}$ , define a sequence of a complex number by $z_{1} = 0, z_{n + 1} = z_{n}^{2} + i for n \geq 1$ .In the complex plane, how far from the origin z₁₁₁ ?

Moderate

A

1
B

$\sqrt{2}$
C

$\sqrt{3}$
D

$\sqrt{110}$

Solution

$\begin{array}{l} z_{1} = 0, z_{n + 1} = z_{n}^{2} + i, n \geq 1 \\ z_{2} = z_{1}^{2} + i at n = 1 \\ z_{2} = 0 + i \Rightarrow z_{2} = i \\ z_{3} = z_{2}^{2} + i \Rightarrow z_{3} = i^{2} + i = - 1 + i \\ z_{4} = z_{3}^{2} + i \Rightarrow z_{4} = (- 1 + i)^{2} + i \\ = 1 + i^{2} - 2 i + i = - i \end{array}$
$\begin{array}{l} z_{5} = z_{4}^{2} + i \\ \Rightarrow z_{5} = (- i)^{2} + i = - 1 + i \end{array}$
Hence, $z_{111} = - 1 + i$

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