Sum of Complex Numbers in Exponential Form/General Result

Theorem
Let $n \in \Z_{>0}$ be a positive integer.

For all $k \in \set {1, 2, \dotsc, n}$, let:
 * $z_k = r_k e^{i \theta_k}$

Let:
 * $r e^{i \theta} = \displaystyle \sum_{k \mathop = 1}^n z_k = z_1 + z_2 + \dotsb + z_k$

Then: