Infinite circuits are easy. How about long ones?

Abstract

We consider a long but finite (ladder) circuit with alternating connections of resistors in series and parallel and derive an explicit expression for its equivalent resistance as a function of the number of repeating blocks, $R_{\rm eq}(k)$. This expression provides an insight on some adjacent topics, such as continued fractions and a druncard's random walk in a street with a gutter. We also remark on a possible method of solving the non-linear recurrent relation between $R_{\rm eq}(k)$ and $R_{\rm eq}(k+1)$. The paper should be accessible to students familiar with the arithmetics of determinants.