Complex number

Upon observation, $\mathbb{Im}(z) = i$ in order to satisfy $|z|$ being a real number. The real part, $a = \mathbb{Re}(z),$ can be computed per:

$|a+i| = \sqrt{a^2 + 1} = 1 + (a+i) - i \Rightarrow a^2 + 1 = a^2 + 2a + 1 \Rightarrow 0 = 2a \Rightarrow a = 0$

so that $z = i$ . The above geometric series finally yields:

$\Sigma_{k=1}^{20} z^{k} = \frac{z(1-z^{20})}{1-z} = \frac{(i)(1 - (i^4)^5)}{1-i} = \frac{(i)(1 - 1^5)}{1-i} = \boxed{0}.$