If z^2 + 1 = 0, find z^(49) + z^(50) + z^(51) + z^(52).
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 \(z^{49}+z^{50}+z^{51}+z^{52}=0\)
