A New Generalization of the Golden Ratio

We propose a generalization of the golden section based on division in mean and extreme ratio. The associated integer sequences have many interesting properties. 1. GENERALIZED GOLDEN RATIOS There have been many generalizations of the number known as golden ratio or golden section, φ = 1+ √ 5 2 . Examples are G.A. Moore’s golden numbers [10] and S. Bradley’s nearly golden sections [5] (see also [7] and [9]). A generalization that has been considered by several authors are the positive roots of x − x − 1 = 0; see [12] and [14]. In this paper, a similar generalization is proposed. It is based on the original definition of φ, division of a line segment in mean and extreme ratio. Let G be a point dividing the segment AB in parts of length a = |AG| and b = |GB|; suppose a > b. The division is mean and extreme if the ratio of the larger to the smaller part equals the ratio of the whole segment to the larger part: