On a new closed formula for the solution of second order linear difference equations and applications

In this paper, we establish a new closed formula for the solution of homogeneous second order linear difference equations with constant coefficients by using only matrix theory. This, in turn, gives new closed formulas for many sequences of this type such as Fibonacci and Lucas sequences and many others. Then we present two applications: one deals with finding interesting summation formulas relating the elements of such sequences. The second application is to show how our approach gives rise to a new method for solving systems of first order difference equations. Finally, we introduce the generalized ratio of such sequences and find a formula for it which can be considered as a generalization of the golden ratio, the silver ratio, and many others.