Special Geometric Sum of $\text{i}$

$\begin{aligned} \text{(1) i}^1 &=\text{i} \\ \text{(2) i}^2 &=-1 \\ \text{(3) i}^3 &=-\text{i} \\ \text{(4) i}^4 &=1 \\ \end{aligned}$ $\because (1)+(2)+(3)+(4)=0, \dfrac{2000}{4}=500$ $\therefore \text{i}+\text{i}^2+\text{i}^3+\cdots+\text{i}^{2000}=0$

$\begin{aligned} z & = i + i^2 + i^3 + \cdots + i^{2000} & \small \color{#3D99F6} \text{It is a geometric sum} \\ & = i \times \frac {{\color{#3D99F6}i^{2000}}-1}{i-1} & \small \color{#3D99F6} \text{Note that }i^4 = 1 \\ & = i \times \frac {{\color{#3D99F6}1^{400}}-1}{i-1} \\ & = \boxed 0 \end{aligned}$