Let i=−1, define a sequence of a complex number by z1=0,zn+1=zn2+i for n≥1.In the complex plane, how far from the origin z111 ?

z1=0,zn+1=zn2+i,n≥1z2=z12+i at n=1z2=0+i⇒z2=iz3=z22+i⇒z3=i2+i=−1+iz4=z32+i⇒z4=(−1+i)2+i=1+i2−2i+i=−i z5=z42+i⇒ z5=(−i)2+i=−1+iHence,z111=−1+i

Let i=−1, define a sequence of a complex number by z1=0,zn+1=zn2+i for n≥1.In the complex plane, how far from the origin z111 ?

z1=0,zn+1=zn2+i,n≥1z2=z12+i at n=1z2=0+i⇒z2=iz3=z22+i⇒z3=i2+i=−1+iz4=z32+i⇒z4=(−1+i)2+i=1+i2−2i+i=−i z5=z42+i⇒ z5=(−i)2+i=−1+iHence,z111=−1+i

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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 ?

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$\sqrt{110}$

$\sqrt{3}$

$\sqrt{2}$

answer is B.

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Detailed Solution

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