A "bug" of negligible dimensions starts at the
origin(0,0)
of the standard two-dimensional rectangular
coordinate system.
The bug walks one unit right, then one-half unit
up, then 1/4 unit
left, then 1/8 unit down, etc.
In each move, it always turns counter-clockwise at
a 90 degree
angle and goes half the distance it went on the
previous move.
Which point (x,y) in the xy-plane is the bug
approaching in its
spiraling journey?
Solution to Problem:
Answer is (4/5, 2/5)
Solution:
You can think of the x and y coordinates as each
being the sum
of an infinite geometric series.
x = 1 - 1/4 + 1/16 - 1/64 + 1/256 + ...
Using the formula for the sum of an infinite
geometric series,
S = a1 / (1 - r) where a1 = 1 (the first term) and r
= -1/4 (common ratio),
S = 1 / (1 - (-1/4)) = 1 / (5/4) = 4/5
Similarly, y = 1/2 - 1/8 + 1/32 - 1/128 + ...
So, S = (1/2) / (1 + 1/4) = 2/5
Correctly solved by:
| 1. Bob Hearn | Winchester, VA |
| 2. Chip Schweikarth | Winchester, VA |
| 3. Jonathan Pence | Virginia Tech |