Geometry or Algebra?
Concentric circles of radii 1,2,3,....100 centimeters are drawn. The interior of smallest circle is colored red and the angular regions are colored alternately green and red, so that no two adjacent regions are of same color. The total area of green regions (in units of square centimeter) is equal to
The answer is 15865.08.
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1 solution
The total area can be found by adding all of the green rings of radius 2,4,6,8 ... all of which have thickness one.
The areas for the green concentric rings are given below, which need to be summed
2nd: 2 2 π - 1 2 π = 3π
4th: 4 2 π - 3 2 π = 7π
6th: 6 2 π - 5 2 π = 11π
8th: 8 2 π - 7 2 π = 15π
....... etc to 100th = 199π
Sum of areas - arithmetic sequence: ( ( 3 + 1 9 9 ) × 2 5 ) π
= 5050π = 15865.08