A Phenomenon in Fibonacci Numbers and Its Generalization

Motivated by the result of Fibonacci numbers for which the ratio of successive terms tends to a limit, which is commonly known as the Golden Ratio, we prove an immediate generalization for a wider class of recurrence sequences. We note that such limiting behavior for ratio of successive terms of general linear recurrence sequences has been well discussed, but still they need to satisfy specic conditions for the limit to exist. Our contribution is that we show that such conditions are indeed satised for the cases we are considering. For an application of our main result, we find a natural way to approximate an algebraic number, which is a zero for some class of polynomial equations, by rational numbers. As recently there seem to be renewed interests on Fibonacci numbers and related recurrence sequences, we hope that our elementary methods and results may shed some light for solving the related problems.