  In this paper we study how to accelerate the convergence of the ratios (x_n)
of generalized Fibonacci sequences. In particular, we provide recurrent
formulas in order to generate subsequences (x_{g_n}) for every linear recurrent
sequence (g_n) of order 2. Using these formulas we prove that some
approximation methods, as secant, Newton, Halley and Householder methods, can
generate subsequences of (x_n). Moreover, interesting properties on Fibonacci
numbers arise as an application. Finally, we apply all the results to the
convergents of a particular continued fraction which represents quadratic
irrationalities.
