Recurrence Relation

The Characteristic Root Technique

Given a recurrence relation $a_n+\alpha a_{n-1}+\beta a_{n-2}=0\text{,}$ the characteristic polynomial is $x^2+\alpha x+\beta$ giving the characteristic equation: $x^2+\alpha x+\beta =0.$ then the solution to the recurrence relation is

• for Distinct roots: $a_n=A{r}_1^n+B{r}_2^n$
• for Repeated roots: $a_n=A{r}^n+Bn{r}^n$

where $A$ and $B$ are constants determined by the initial conditions of a recurrence relation.

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$r=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$