Problem:
A bug (of negligible size) starts at the origin on the co-ordinate plane. First it moves 1 unit right to (1,0). Then it makes a 90∘ turn counterclockwise and travels 21 a unit to (1,21). If it continues in this fashion, each time making a 90∘ turn counterclockwise and traveling half as far as in the previous move, to which of the following points will it come closest?
Answer Choices:
A. (32,32)
B. (54,52)
C. (32,54)
D. (32,31)
E. (52,54)
Solution:
If the bug travels indefinitely, the algabraic sum of the horizontal components of its moves approaches 54, the limit of the geometric series
1−41+161−⋯=1−(−41)1
Similarly, the algebraic sum of the vertical components of its moves approaches 52=21−81+321⋯. Therefore, the bug will get arbitrarily close to (54,52).
OR
The line segments may be regarded as a complex geometric sequence with a1=1 and r=i/2. Thus
i=1∑∞ai=1−ra1=2−i2=54+2i.
In coordinate language, the limit is the point (54,52).