Let Z0 = I, Z1 = 1+i, and for n ≥ 2 define Zn = Zn−1Zn−2. Prove that, for every non-negative integer n, Re(zn) Im(zn) = 0 if and only if 3 | n.

 

Z₀ = i

Z₁ = 1 + i

for n ≥ 2 , Z_n = Z_n-1 Z_n-2

we substitute n =2 to Z_n = Z_n-1 Z_n-2;

Z₂ = Z_2-1 Z_2-2

Z₂ = Z₁ Z₀

= [1+i] [i]

Z₂ = i - 1

now substitute n= 3;

Z₃ = Z_3-1 Z_3-2

= Z₂ Z₁

= [i -1 ] [i +1]

= i² - 1 = -2

Z₃ = -2

For n=6, we can get Z₆ = 16i

Therefore, for any divisible by 3, it has either only real part or only imaginary part.

In conclusion, R_e(Z_n) I_m(Z_n) = 0 if and only if 3 | n, as for any other n. Both imaginary and real parts exist.