Answer: Infinite Resistors

Remember that the problem was to find the equivalent resistance of this thing with R_1 = R_2 = 1:

.

We know the equivalent resistance must exist, because it can’t be less than zero, and every time we add more resistors in series the resistance decreases, so we have a bounded monotonic series.

Replacing the back part of the circuit with its equivalent resistance, we get a diagram like this:

resistors

Now find the equivalent resistance to x.

x = 1 + \left(\frac{1* x}{1 + x}\right)

which simplifies to

x = \frac{1 + \sqrt{5}}{2} ,

which is the golden ratio.

One Response to “Answer: Infinite Resistors”

  1. Infinite Resistors 7/17/11 « Kappa Alpha Theta Summer 2011 Says:

    […] This is a famous problem, solved many places online. I wrote a solution here. […]

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