An Analytic Approximation of the Golden Ratio via a 24th-Root and Exponential Tail

Abstract

The golden ratio, denoted by φ, is one of the most fundamental mathematical constants, with deep connections to number theory, geometry, natural sciences, and aesthetic structures. In this article, Dr. Fazal Rehman and Dhan Bir Limbu present a novel analytic approximation of the golden ratio based on a hybrid formulation involving a high-order root and an exponential–factorial series tail. The proposed expression represents φ as the sum of the twenty-fourth root of a fixed integer and the residual terms of an exponential series beginning from a higher-order index. A detailed step-by-step evaluation demonstrates that this formulation converges rapidly and produces a numerical value that closely matches the accepted value of the golden ratio to high precision. This work introduces a new analytical pathway for approximating φ and highlights the effectiveness of exponential tail corrections in refining root-based approximations of fundamental mathematical constants.