Brilli's Spiral Journey

We can consider Brilli to be on the complex plane. We can then define Brilli's kth movement (starting with the 0th movement) to be $(\frac{1}{2}e^{i\frac{\pi}{3}})^k$ .

To find point $M$ , we obtain the infinite series: $\sum\limits_{k=0}^{\infty}(\frac{1}{2}e^{i\frac{\pi}{3}})^k$

The absolute value of this ratio is less than 1, therefore we can use the formula for an infinite geometric series, $\frac{1}{1-r}$ :

$\sum\limits_{k=0}^{\infty}(\frac{1}{2}e^{i\frac{\pi}{3}})^k=\frac{1}{1-\frac{1}{2}e^{i\frac{\pi}{3}}}$

This evaluates to $1+i\frac{\sqrt{3}}{3}$ . The absolute value of this number is $\sqrt{\frac{4}{3}}$ , therefore $a=4$ , $b=3$ , and $a+b=\boxed{7}$ .

A video for the solution. It is more interesting if it generalized to 3-D.