homework and exercises - Equivalent resistance in ladder circuit

I have stumbled upon a given question I really have a hard time to solve. Basically I need to find an equivalent resistance in some form of "ladder" configuration. Where the chain is an infinite sequence of resistors.

I have really no good idea how to find this equivalent resistance. Trying the old fashioned rule of parallel and resistors in series I came to a very messy formula:

$$R + \frac{1}{\frac{1}{R} + \frac{1}{R+\frac{1}{\frac{1}{R}+....}}}$$

I know however that the solution should be much more simple.

Now I tried using Kirchhoff's loop rule. Which states that the power difference in a closed loop must be 0. Naming the "potential currents" between AB $I_1$, between BC (through the single resistance) $I_2$ and the current "from B to the right" $I_3$ Considering the loop containing BC & the rest of the structure this rewrites to:

$$I_1 = I_2 + I_3$$ $$I_3 \cdot R - I_2 \cdot R_{eq} = 0$$

The problem is, 2 variables, 2 functions doesn't really bring me closer to an answer :(. What am I missing?

Answer

HINT:

Notice that

$$R_{eq}=R+\frac1{\frac1R+\frac1{R_{eq}}}$$