Summing Imaginary Powers

Let $\displaystyle S(n) = \sum_{k=1}^n ki^k = i - 2 - 3i + 4 +5i - 6 -7i + 8 + 9i - ... + ni^n$

Note that:

$\begin{aligned} S(1) & = 0 + i \\ S(5) & = 2 + 3 i \\ S(9) & = 4 + 5 i \\ ... \ & = \ ... \\ \implies S(n) & = \frac {n-1}2 + \frac {n+1}2i & \small \color{#3D99F6}{\text{where } n \text{ is odd.}} \\ & = 1280+1281i \end{aligned}$

$\begin{aligned} \implies \frac {n-1}2 & = 1280 \\ n-1 & = 2560 \\ \implies n & = \boxed{2561} \end{aligned}$