A Curious Property of the Eleventh Fibonacci Number
Benjamin M. M. de Weger · Rocky Mountain Journal of Mathematics · 1995
Introduction As usual we denote by F n the nth Fibonacci number, defined recursively by F 0 = 0; F 1 = 1 and F n = F n\\Gamma1 + F n\\Gamma2 for all n 2. The decimal expansion of the reciprocal of the eleventh Fibonacci number F 11 = 89 has a remarkable shape: its 6 leading digits are the first 6 terms of the Fibonacci sequence, viz. 1 89 = 0:011235955 : : : Looking more closely, it becomes apparent that the relation goes even beyond the 6th decimal place: 1 89 = 0 10 1 + 1 10 2 + 1 10 3 + 2 10 4 + 3 10 5 + 5 10 6 + 8 10 7 + 13 10 8 + 21 10 9 + 34 10