**Problem 3.152 of Irodov**

The figure below shows an inifite circuit formed by the repetition of the same link, consisting of resistance and . Find the resistance of this circuit between points and .

**Solution**

Let’s denote the resistance between the points and by . Since the circuit is infinite, removing the first and resistors gives the same arrangement back again — the arrangement is *self-similar*. That means, the resistance between the points and is just without the left-most and resistors, and we may redraw the circuit as shown below.

It is now straightforward to calculate the resistance,

(1)

which gives a quadratic equation for ,

(2)

This equation has two roots. We keep only the positive root, because resistance cannot be a negative number.

(3)

Plugging in and , we find .

**Bonus Problem:** At what value of the resistance in the circuit shown below will the total resistance between points and be independent of the number of cells. (*Hint:* What value of will make the circuit appear self-similar at the lower rungs of the ladder?)

Answer: .