# Infinite Number of Circles

Geometry Level 5

On a straight line $$\ell$$, we have an infinite sequence of circles $$\Gamma_n$$, each with radius $$\frac {1}{2^n}$$, such that $$\Gamma_n$$ is externally tangential to the circles $$\Gamma_{n-1}, \Gamma_{n+1}$$ and the line $$\ell$$. Consider another infinite sequence of circles $$C_n$$, each with radius $$r_n$$, such that $$C_n$$ is externally tangential to $$\Gamma_{n}, \Gamma_{n+1}$$ and $$\ell$$, and contained within region bounded by them. The expression $$\displaystyle \sum_{i=1}^\infty r_i$$ can be expressed as $$a - \sqrt{b}$$, where $$a$$ and $$b$$ are positive integers. What is the value of $$a+b$$?

This problem was proposed by Arunatpal.

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