This is no ordinary Fibonacci spiral

Let the starting point of the race the city center be the origin $O(0,0)$ of the complex plane. Let east and west direction be the positive and negative real axis respectively, and north and south, the positive and negative imaginary axis respectively. Let each straight run or side length with direction be denoted by complex number $a_n$ , where $n$ is a positive integer. Then $a_1 = 1$ , $a_2 = \frac 12 i$ , $a_3 = - \frac 13$ , $a_4 = - \frac 15 i$ , ... $a_n = \frac {i^{n-1}}{F_n}$ .

Let $\displaystyle s_n = \sum_{k=1}^n a_k$ . We note that |s_n| is the distance between origin and the end of $n$ th side. Then, we have:

$\begin{aligned} \lim_{n \to \infty} s_n & = \lim_{n \to \infty} \sum_{k=1}^n \frac {i^{k-1}}{F_k} \\ & = \sum_{k=0}^\infty \frac {(-1)^k}{F_{2k+1}} + i \sum_{k=1}^\infty \frac {(-1)^{k+1}}{F_{2k}} & \small \color{#3D99F6} \text{By numerical method} \\ & = 0.644358338 + 0.757204375i & \small \color{#3D99F6} \text{ (in hectometers)} \end{aligned}$

Now $l = 100 |s_\infty| = 100 \times 0.994261602$ m, $\implies \lfloor l \rfloor = \boxed{99}$ m.