Question

[Key Concept: An infinite ladder looks the same after removing its first cell, so its input resistance satisfies a self-consistency equation.]

$$\begin{matrix} & \frac{R_{2}x}{R_{2} + x} + 2R_{1} = x \\ \Rightarrow & R_{2}x + 2R_{1}R_{2} + 2R_{1}x = R_{2}x + x^{2} \\ \Rightarrow & x^{2} - 2R_{1}x - 2R_{1}R_{2} = 0 \\ \Rightarrow & x = \frac{2R_{1} \pm \sqrt{4R_{1}^{2} + 8R_{1}R_{2}}}{2} \\ & x = R_{1} + \sqrt{R_{1}^{2} + 2R_{1}R_{2}} \end{matrix}$$