Let `i=sqrt(-1)` Define a sequence of complex number by `z_1=0, z_(n+1) = (z_n)^2 + i` for `n>=1`. In the complex plane, how far from the origin is `z_111`?

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

B

Explanation

`(b)` `z_(1)=0`, `z_(n+1)=z_(n)^(2)+i`
Put `n=1impliesz_(2)=z_(1)^(2)+i=i`

Put `n=2impliesz_(3)=z_(2)^(2)+i=i^(2)+i=i-1`
Put `n=3impliesz_(4)=z_(3)^(2)+i=(i-1)^(2)+i=-i`
Put `n=4impliesz_(5)=z_(4)^(2)+i=(-i)^(2)+i=i-1`
Put `n=5impliesz_(6)=z_(5)^(2)+i=(i-1)^(2)+i=-i`
Thus `z_(111)=-1+i`