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​).