Ant Walking in a Squared Spiral

Date: 06/02/99 at 02:17:01
From: Kevin Marnell
Subject: Using limits to determine coordinates

My math teacher gave me this question:

An ant of negligible size walks out a distance of 1 from the origin, 
down the x-axis. It then turns left and goes up 1/2 from its current 
point. If the ant continues turning left, going the half the distance 
it previously went, and repeating the pattern, where does the ant 
eventually end up?

I found that the side-to-side motion along the x-axis follows the 
pattern:  1, -1/4, 1/16/, -1/64 or 1/[-4^(n-1)]

The y axis movement goes:  1/2, -1/8, 1/32, -1/128

I've noticed that the point is close to (3/4, 7/8). I have no idea how 
to find the exact point. Can you help me?

Date: 06/02/99 at 11:54:36
From: Doctor Rob
Subject: Re: Using limits to determine coordinates

The sum of all the x-direction motions is a geometric series, that is, 
of the form

   a + a*r + a*r^2 + a*r^3 + ... + a*r^(n-1) + ...

Figure out what a and r are, then apply the formula for the sum of a 
geometric series.

A similar idea works for the y-direction motions.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/