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[l]} and minimal extra-super increasing sequence {z[0], ...,[z]l}. Discover 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 able to be expressed by computers. 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 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.

中文翻译：

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5 的平方根的一种新的有理方法

在本文中，作者构造了一种称为超超增序列的新型序列，并给出了最小超增序列{a[0], ..., a[l]}和最小超增序列的定义。超级递增序列 {z[0], ...,[z]l}。发现总是存在一个使 (z[n] / z[n-1] - a[n] / a[n-1])= PHI 的拟合 n，其中 PHI 是具有有限精度的黄金比例共轭用计算机来表达。此外，推导出公式 radic(5) = 2(z[n] / z[n-1] - a[n] / a[n-1]) + 1，其中 n 对应于所需的精度。实验表明，通过比率差来获得 radic(5) 的方法比通过泰勒幂级数更平滑和迅速，并说服作者 lim{n 到无穷大} (z[n] / z[n-1] - a[n] / a[n-1]) = PHI 成立。