Triangle Circuit

Electricity and Magnetism Level 4

A circuit is built by connecting \(1\ \Omega\) wires in a pattern of triangles, as shown above. Point \(A\) is the common vertex of all triangles, and point \(B\) is a vertex of one of the triangles at the end of the pattern.

Let \(n\) be the number of triangles in the pattern. (In the diagram above, \(n = 7\).)

Determine the resistance between points \(A\) and \(B\) in the limit where the pattern of wires grows to infinitely many triangles, i.e. \[\lim_{n\to\infty} R_{AB}\]

Give your answer to 3 decimal places.


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