Ant Walking in a Squared SpiralDate: 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/ |
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