What Coordinates Does the Ant Reach in Its Spiral Path?

[Macleef]

Homework Statement

An ant of negligible dimensions start at the origin (0,0) of the standard 2-dimensional rectangular coordinate system. The ant walks one unit right, then one-half unit up, then one-quarter unit left, then one-eighth unit down, etc. In each move, it always turn counter-clockwise at a 90 degree angle and goes half the distance it went on the previous move. Which point (x,y) in the xy-plane in the ant approaching in its spiraling journey?

Homework Equations

I think you use the geometric series to solve this problem?

The Attempt at a Solution

I don't have an attempt at this problem because I don't know where to begin!
I don't know how to solve this problem! All I know is you use the geometric series??
And if you do, how would you go solve this problem with the geometric series?


The answer is: (4/5 , 2/5)

 

[Dick]

Write down a series of all of the x displacements and another series of all the y displacements. They should be geometric series. Then you can start worrying about summing them.

 

[chaoseverlasting]

The sum will be unto infinity.

 

[HallsofIvy]

It appears to be ChaosEverlasting's goal to spread everlasting chaos!

As Dick suggested, look at "x" (East,West) and "y" (North, South) components separately. That's easy since the ant alternates between going East-West and North-South.

Yes, as ChaosEverlasting implies, you will have two infinite series. However, since they are alternating series (positive, then negative), both series converge. In fact, they are simple geometric series.

 

[Diffy]

Do you know how to find the sum of an infinite geometric series?

 

[Gib Z]

If I was an Ant, I'd start swinging at you. "Negligible Dimensions", pfft. =]

 

[HallsofIvy]

Yeah, ants tend to be really sensitive about their size!