On the recursive sequence 𝑥_{𝑛+1}=\frac{𝐴}𝑥_{𝑛}+\frac{1}𝑥_{𝑛-2}

Abstract

We show that every positive solution of the equation \[ x n + 1 = A x n + 1 x n − 2 , n = 0 , 1 , … , x_{n+1} = \frac {A}{x_{n}} + \frac {1}{x_{n-2}}, \hspace {.2in} n = 0, 1, \ldots , \] where A ∈ ( 0 , ∞ ) A \in (0, \infty ) , converges to a period two solution.