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.

翻译：在本文中,作者构建了新型序列,称为超超增序,并给出了最小超增序 {a[0],...,[l]}和最小超增序 {z[0],...,[z]}。发现始终存在一个匹配的 n,使(z[n] /z[n-1] - a[n-1]/a[n-1])= PHI(PHI),其中PHI是金比方,具有一定精度,计算机可以表达。此外,还得出公式 Radic(5) = 2(z[n] / z[n-1] - a[n] / a[n-1]) + 1,其中n与要求的精度相对应。实验表明,通过比例差使(z[n] /z[n-1] - a[n-1] = PHI。