A New Rational Approach to the Square Root of 5

TL;DR

This paper introduces a novel sequence-based method to approximate the square root of 5 using ratios related to super increasing sequences, demonstrating faster convergence than traditional Taylor series methods.

Contribution

The authors define new types of sequences and establish a formula linking their ratios to the square root of 5, providing a more efficient approximation approach.

Findings

Sequence ratios converge to the golden ratio conjugate.

The method yields a smooth and rapid approximation of √5.

The approach outperforms Taylor series in convergence speed.

Abstract

In this paper, authors construct a new type of sequence which is named an extra-super increasing sequence, and give the definitions of the minimal super increasing sequence {a[0], a[1], ..., a[n]} and minimal extra-super increasing sequence {z[0], z[1], ..., z[n]}. Find that there always exists a fit n which makes (z[n] / z[n-1] - a[n] / a[n-1])= PHI, where PHI is the golden ratio conjugate with a finite precision in the range of computer expression. Further, derive the formula radic(5) = 2(z[n] / z[n-1] - a[n] / a[n-1]) + 1, where n corresponds to the demanded precision. Experiments demonstrate that the approach to radic(5) through a term ratio difference is more smooth and expeditious than through a Taylor power series, and convince the authors that lim(n to infinity) (z[n] / z[n-1] - a[n] / a[n-1]) = PHI holds.