One approach to solving problems involving an infinite system
like this one is to focus on the idea of infinity and its
properties. Removing one item from an infinite system does not
change the value for the whole system. This idea is the key to the
simplest method of solving this particular problem.
In this diagram the infinite ladder extends downwards and the
network below C and D is effectively the same as the network below
A and B. The total resistance $r_1$ between C and D, for the
infinite network, will be the same as the total resistance $R$
between A and B. So we can think of the resistance between A and B
as a resistance of 1 ohm in parallel with three resistances in
series. Write down the equation and you will get a simple quadratic
equation to solve for $R$.
