complex roots

Hi,
Can someone please explain why the function there represents the movement of the ant?/ how to approach the question. I thought the movement would be some sort of a staircase shape. Thanks

The path represents a section of a regular polygon. So the exterior angle is 2pi/9 and the length is 1. So e^(i 2pi/9) represents a rotation (exterior angle) and +1 represents a step in that direction. The magnitude will represent the length from the origin. Why not write down the first step, second step, third step, ...., nth/4th step ... and you should notice a series.

Note the easiest way to do the question is probably do sketch the polygon and do some simple trig to get the length of the chord. There are a couple of ways you could go.