On the golden number and Fibonacci type sequences

Abstract

The paper presents, among others, the golden number $\varphi$ as the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping of a certain interval with the help of Edelstein's theorem. To demonstrate the equality  , where $f_n$ is $n$-th Fibonacci number also the formula from Corollary \ref{cor1} has been applied. It was obtained using some relationships between Fibonacci and Lucas numbers, which were previously justified.