Complex Numbers

We have $z^{49}+z^{50}+z^{51}+z^{52}=z^{49}(1+z+z^2+z^3)$

The term in parentheses factors as $1+z+z^2+z^3=(z+1)(z^2+1)$

So, if $z^2+1=0$ then $z^{49}+z^{50}+z^{51}+z^{52}=0$