Given a linear homogeneous recurrence of the form , with constant coefficients , and initial conditions , has a characteristic equation given by the formula , this equation has a degree 2 and has two roots and . If then is a solution of the recurrence relation, where and are the solution of the system
From the problem we know that and are the inicial conditions and and . In order to find the general formula for , first we find its characteristic equation
⇔⇔⇔ ∨
Now we need to find and using the initial conditions and , we need to solve the system of equations
you can add the two equations and find the values and , so a general formula for is .