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.
by
Arjen Vreugdenhil