Abstract
In this paper we study the so-called generalized Fibonacci sequence: $x_{n+2} = \alpha x_{n+1} + \beta x_n, n\in \mathbb{N}$. We derive an open domain around the origin of the parameter space where the sequence converges to $0$. The limiting behavior on the boundary of this domain are: convergence to a nontrivial limit, $k$-periodic ($k\in \mathbb{N}$), or quasi-periodic. We use the ratio of two consecutive terms of the sequence to construct a rational approximation for algebraic numbers of the form: $\sqrt{r}, r\in \mathbb{Q}$. Using a similar idea, we extend this to higher dimension to construct a rational approximation for $\sqrt[3]{ a + b\sqrt{c}} + \sqrt[3]{ a - b\sqrt{c}} + d$.
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