| A Sequence of Circles |
I have an infinite number of circles. The first one has radius r. The second has radius r/2. The third has radius r/4, and so on.
I also have an infinite number of squares, each of which has area equal to the area of the corresponding circle.
What is the sum of the perimeters of all my squares? |
Problem Moderated by: Douglas |
| Problem Solution |
The area of the nth circle is
Since each square has the same area as the corresponding circle, and the area of a square is s2
And the perimeter is 4s, or
Therefore, the sum of the perimeters is:
Which is the same as
4√π r (1 + 1/2 + 1/4 + 1/8+...)
This equals 4√π r x 2 = 8√π r
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