Some relations involving the sums of Fibonacci numbers
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In the first section, we give some preliminaries from Fibonacci numbers which are the sequence given by F0 = 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 [수식] as the limit.
Abstract
In the first section, we give some preliminaries from Fibonacci numbers which are the sequence given by F0 = 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 [수식] as the limit.
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