1980 AHSME Problems/Problem 13
Contents
[hide]Problem
A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to . Then it makes a counterclockwise and travels a unit to . If it continues in this fashion, each time making a degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?
Solution
Writing out the change in coordinates and then in coordinates gives the infinite sum and respectively. Using the infinite geometric sum formula, we have and , thus the answer is
Solution
We can represent the bug's position on the coordinate plane using complex numbers. The first move the bug makes is , the second , the third , and so on. It becomes clear that the distance the bug travels is an infinite geometric series with initial term 1, and common ratio .
Thus, applying the infinite geometric series formula:
This is equivalent to the coordinate being and the coordinate being , so the answer is
~ jaspersun
See also
1980 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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