TITLE: Fun Geometric Series Puzzle
QUESTION [42 upvotes]: 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:
$$\frac{a}{\sqrt{r^2 + 1}}$$

REPLY [2 votes]: Another, but equivalent, way of viewing this.
The first two steps together with the line joining their endpoints form a right angled triangle T with hypotenuse h = $a\sqrt(1+r^2)$. 
The third and fourth steps are each $r^2$ of the length of the original sides so they form two sides of a similar triangle with the hypotenuse colinear with the hypotenuse of T.  By similarity, it has length $r^2$h.  Therefore this vertex is $h(1-r^2)$ from your starting point.  This pattern repeats so the vertex of successive triangles all lie on the same line, with distances based on the partial sums of $h(1 - r^2 + r^4 - ...)$.  Therefore the limit, if there is one, must be $h/(1 + r^2) = a/\sqrt(1 + r^2)$.  In this case it's geometrically obvious that the limit does exist, even without using any of the formal methods of confirming it.