Some relations involving the sums of Fibonacci numbers

Abstract

In the first section, we give some preliminaries from Fibonacci numbers which are the sequence given by Fo = 0, F1 = 1 and Fn = Fn-1 + Fn-2 for all n ≥ 2. In the second section, we derive explicit formulas for the Fibonacci numbers Fn F2n, and F2n+1. In the fourth section, we give another method for the sums of Fibonacci numbers using the sums of powers of α and β which are the roots of the characteristic equation of Fn = Fn-1 + Fn-2. In the last section, we consider the cross-ratio for four consecutive Fibonacci numbers Fn, Fn+1, Fn+2 and Fn+3. We proved that the cross-ratio of Fn, Fn+1, Fn+2 and Fn+3 converges to -1/2α as the limit.