Let i= `sqrt(-1)`. Define a sequence of complex number by `z_1=0 , z_(n+1)=z_n^2+1` for n `>=` 1. In the complex plane, then arg(`z_(111)`) is :
| A. |
`pi/4` |
| B. |
`-pi/4` |
| C. |
`(3pi)/4` |
 |
| D. |
`pi/2` |
`z_1 = 0, z_(n+1)= z_n^2+i`
Put n = `1 =>z_2=z_1^2+i=i`
Put n=`2 =>z_3 =z_2^2+i= i^2 + i = i – 1`
Put n = `3=>z_4 = z_3^2+i = (i–1)^2+ i = –i`
Put n =` 4 =>z_5=z_4^2+ i = (–i)^2 + i = i –1`
Put n = `5 =>z_6 = z_6 ^2 + i = (i–1)^2 + i = – i`
Thus arg`(z_(111))` = arg(i-1) = `(3pi)/4`