Originally Posted by

**puggie** I need help to solve this question.

An ant of negligible size walks a distance of 10 units 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 co-ordinates of the point where the ant will eventually end up. Answer: ( 8, 4 )

Hi **puggie**.

Consider its horizontal and vertical perambulations separately. Horizontally, its displacements are

$\displaystyle 10\,-\,10\left(\frac14\right)\,+\,10\left(\frac14\right )^2\,-\,10\left(\frac14\right)^3\,+\,\cdots$

That is, 10 units to the right, $\displaystyle 10\left(\frac14\right)$ units to the left, $\displaystyle 10\left(\frac14\right)^2$ units to the right, $\displaystyle 10\left(\frac14\right)^3$ to the left, and so on.

Vertically, it moves

$\displaystyle 5\,-\,5\left(\frac14\right)\,+\,5\left(\frac14\right)^ 2\,-\,5\left(\frac14\right)^3\,+\,\cdots$

That is, 5 units up, $\displaystyle 5\left(\frac14\right)$ units down, $\displaystyle 5\left(\frac14\right)^2$ units up, $\displaystyle 5\left(\frac14\right)^3$ units down, and so on.

Now sum these two infinite geometric series.