On a straight line ℓ, we have an infinite sequence of circles Γn, each with radius 1/2^n, such that Γn is externally tangential to the circles Γn−1,Γn+1 and the line ℓ. Consider another infinite sequence of circles Cn, each with radius rn, such that Cn is externally tangential to Γn,Γn+1 and ℓ. The expression ∑i=1 to ∞ ri can be expressed as a−√b, where a and b are positive integers. What is the value of a+b?

Question

On a straight line ℓ, we have an infinite sequence of circles Γn, each with radius 1/2^n, such that Γn is externally tangential to the circles   Γn−1,Γn+1 and the line ℓ. Consider another infinite sequence of circles Cn, each with radius rn, such that Cn is externally tangential to Γn,Γn+1 and ℓ. The expression ∑i=1 to ∞ ri can be expressed as a−√b, where a and b are positive integers. What is the value of a+b?

Clarification: In this problem, we have a row of circles placed on a line. All points of tangency are distinct. The circle Cn is uniquely determined

harsh · Answer

okay at least somebody give a hint