# Thread: Geometric progression: An ant

1. ## Geometric progression: An ant

An ant of negligible size walks a distance of 10 units from the origin in the x-y plane along the x-axis. It then turns left and goes up 5 units from its current point. If the ant continues turning left and going half the distance it had previously walked, repeating the pattern, find the coordinates of the point where the ant will eventually end up.

2. Hint: After each four turns, the ant is $\displaystyle \frac{3}{4}$ of a Unit (in this case your Unit is 10 units) to the right of where it started, and $\displaystyle \frac{3}{8}$ of a Unit above where it started.

3. Originally Posted by Prove It
Hint: After each four turns, the ant is $\displaystyle \frac{3}{4}$ of a Unit (in this case your Unit is 10 units) to the right of where it started, and $\displaystyle \frac{3}{8}$ of a Unit above where it started.
sorry but i didnt quite understand how i should use the hint

4. Hello, Punch!

An ant of negligible size walks a distance of 10 units from the origin in the x-y plane
along the x-axis. It then turns left and goes up 5 units from its current point.
If the ant continues turning left and going half the distance it had previously walked,
find the coordinates of the point where the ant will eventually end up.

$\text{Note: }1 - \frac{1}{2^2} + \frac{1}{2^4} + \frac{1}{2^6} - \hdots \;\text{is a geometric series with }a = 1\text{ and }r = \text{-}\tfrac{1}{4}$
. . . . . . . . $\text{Its sum is: }\:S \;=\;\frac{1}{1-\left(-\frac{1}{4}\right)} \;=\;\frac{1}{\frac{5}{4}} \:=\:\tfrac{4}{5}$

Sketch the ant's path for several phases.
We note a pattern in the horizontal and vertical displacements.

$x \;=\;10 -\tfrac{10}{2^2} + \tfrac{10}{2^4} - \tfrac{10}{2^6} + \hdots \;=\;10\left(1 - \tfrac{1}{2^2} + \tfrac{1}{2^4} - \tfrac{1}{2^6} + \hdots\right) \;=\;10\left(\tfrac{4}{5}\right) \;=\;8$

$y \;=\;\tfrac{10}{2} - \tfrac{10}{2^3} + \tfrac{10}{2^5} - \hdots \;=\;\tfrac{10}{2}\left(1 - \tfrac{1}{2^2} + \tfrac{1}{2^4} - \hdots\right) \;=\;5\left(\tfrac{4}{5}\right) \;=\;4$

$\text{The ant will eventually end up at }(8,\,4).$