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A "Cartesian Bug"

Source:

December 31, 2005
analytic geometrygeometric series

Problem Statement

A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to (1,0)(1,0). Then it makes a 90∘90^\circ counterclockwise and travels 12\frac 12 a unit to (1,12)\left(1, \frac 12 \right). If it continues in this fashion, each time making a 90∘90^\circ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest? (A) (23,23)(B) (45,25)(C) (23,45)(D) (23,13)(E) (25,45)\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)
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