I recently was reminded of a puzzle I solved in college and thought I'd give it a shot again. However, being distanced from college math, I am having a harder time remembering how I arrived at the solution.
The problem is as follows: Imagine you are standing in the middle of an open field. You walk forward 16 feet, turn right and walk 8 feet, turn right and walk 4 feet, and so on. This continues indefinitely. When you finally reach an infinite number of turns, how far will you be from your original starting point (as the crow flies)? Generalized, here is what the problem looks like:
I did manage to find the original solution that I came up with for the problem. However, I did not show my work so the process is lost. After attempting to resolve this without any luck, I thought I would toss this out to the community to solve as a fun puzzle.
For reference, this is what I believe to be the solution:
$$\frac{a}{\sqrt{r^2 + 1}}$$
Best Answer
We can describe this walk in the complex plane. Starting at $z=0$ we add in sequence $$a,-i r a, -r^2 a, ir^3 a,\cdots, (-i r)^k a,\cdots$$ This is simply the geometric series $$\sum_{k=0}^\infty (-i r)^k a = \frac a{1+ir}$$ which is at a distance of $$\left|\frac a{1+ir}\right|=\frac a{\sqrt{1+r^2}}$$ from the origin.
Additionally :
An immediate generalization is that if we turn an angle of $\phi$ to the right rather than $\dfrac\pi2$, we have the series $$\sum_{k=0}^\infty a\ (r e^{-i\phi})^k = \frac a{1-r e^{-i\phi}}$$ and thus a displacement of $$\frac a{\sqrt{1-2r\cos\phi+r^2}}$$ from the origin.