Connecting Greek Ladders and Continued Fractions

Overview

Greek ladders and continued fractions are both techniques for approximating the \(n\)th root of \(k\) by rational numbers where \(n\) and \(k\) are positive integers. That is, both provide techniques to construct sequences of rationals that converge to \(\sqrt[n]{k}\). In both cases, the sequences are defined recursively but the recursions involved are quite different. In this investigation we will show how to construct a continued fraction that produces the same sequence of approximations as the “classic” Greek ladder for \(\sqrt{k}\). The proof techniques will involve a use of proof by induction that is more subtle than in the usual examples seen by students learning induction for the first time. Throughout this investigation, we will present the reader opportunities to further investigate questions related to Greek ladders and continued fractions.