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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