It seems to be fairly straight-forward . . .
One hundred concentric circles with radii 1, 2, 3, …, 100 are drawn in a plane. The interior
of the circle of radius one is colored red, and each region bounded by consecutive circles
is colored either red or green, with no two adjacent regions the same color.
What is the ratio of the total area of the green regions to the area of the largest circle?
The total area of the green regions is: .
We have an arithmetic series with first term ,
. . last term and terms.
Its sum is: .
Hence, the total area of the green regions is: .
The area of the largest circle is: .
Therefore, the ratio is: .