The inner circle

Geometry Level 3

A circle of radius \(a\) is drawn tangent to the \(x\) and \(y\) axes. This is circle #1, shown above as the largest, red circle.

Circle #2 is a smaller circle drawn tangent to the \(x\) and \(y\) axes and to circle #1.

Circle #3 is an even smaller circle tangent to the \(x\) and \(y\) axes, and to circle #2.

In general, circle #N is drawn tangent to the \(x\) and \(y\) axes and to circle #(N-1).

Which decimal is closest to the ratio of the area of circle #101 to the area of circle #100?

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