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?

Question

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?

Answer:
(4/5 , 2/5)

How do you get this answer? I'd really appreciate it if you explain me how to get that answer!

Thanks

Answer 1

Hint: Write the formula for its movements in the x direction only.

Also write the formula for its movements in the y direction.

Both of your equations will have limits.

Your coordinate solution will be the limits as the ant moves an infinite number of times.

Answer 2

Can you please explain how to do this question thoroughly? I don't know what you're talking about...

Answer 3

To find the point (x, y) that the ant is approaching in its spiraling journey, we can break down the steps taken by the ant and determine the pattern.

1. The ant starts at the origin (0,0).
2. The ant walks one unit to the right, reaching the point (1,0).
3. The ant then walks one-half unit up, reaching the point (1,1/2).
4. The ant walks one-quarter unit to the left, reaching the point (3/4, 1/2).
5. Next, the ant walks one-eighth unit down, reaching the point (3/4, -1/8).

We can notice a pattern in the movement of the ant:
- The x-coordinate alternates between moving positive and negative, with each step halving the distance moved.
- The y-coordinate doubles the distance moved in the positive direction, then halves the distance in the opposite direction.

Let's continue with these observations:

6. The ant walks one-sixteenth unit to the right from the previous point, reaching the point (7/8, -1/8).
7. The ant walks one-thirty-second unit up, reaching the point (7/8, 1/16).
8. The ant walks one-sixty-fourth unit to the left, reaching the point (15/16, 1/16).
9. The ant walks one-hundred-twenty-eighth unit down, reaching the point (15/16, -1/128).

We can observe that with each step, the denominator in the fraction doubles. So, based on this pattern, we can conclude that after n steps, the denominator will be 2^n, and the numerator will be (2^n) - 1.

Using this pattern, we can determine the coordinates of the point that the ant is getting closer to by taking a large number of steps. Let's use n = 7 as an example:

10. The ant walks one-hundred-twenty-eighth (1/2^7) unit to the right from the previous point, reaching the point (255/256, -1/128).
11. The ant walks one-two-hundred-fifty-sixth (1/2^8) unit up, reaching the point (255/256, 1/256).
12. The ant walks one-five-hundred-twelfth (1/2^9) unit to the left, reaching the point (511/512, 1/256).
13. The ant walks one-thousand-twenty-fourth (1/2^10) unit down, reaching the point (511/512, -1/1024).

By taking many more steps following the same pattern, we can observe that the x-coordinate approaches 1, and the y-coordinate approaches 0. This means that the point the ant is approaching is (1, 0).

Therefore, the answer provided in the question, (4/5, 2/5), is incorrect. The correct answer is (1, 0).