# Infinite triangular Resistance

**Electricity and Magnetism**Level 3

Determine the resistance \({ R }_{ AB }\) between the points A and B of the frame made of thin homogeneous wire (as shown in figure), assuming that the number of successively embedded equilateral triangles (with sides decreasing by half) tends to infinity. Side \(AB\) is equal to \(a\) units and the resistance of unit length of wire is \(\rho\) units.

If the resistance \({ R }_{ AB }\) is of the form \(a\rho \left( \dfrac { \sqrt { d } -c }{ f } \right) \), find \(d+c+f\).