A [[Linear Transformation|Linear]] homogeneous [[Recurrence Relation]] of degree $k$ with constant coefficients is a [[Recurrence Relation]] of the form: $\huge a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k} $ $\large \forall n : c_{n} \neq 0$. Where none of the coeffiences are 0, Homogeneous in this context referes to how all terms must be multiples of a previous element in the [[Sequence]]. >[!tldr] Thereom: > >Let $c_{1},c_{2}\in\R$. Suppose that $r^2-c_{1}r-c_{2}=0$ has distinct solutions $r_{1},r_{2}$. Then the [[Sequence]] $\set{a_{n}}$ is a solution to the [[Recurrence Relation]]. > >$\large >\begin{align} > >a_{n} = c_{n}a_{n-1} + c_{2}a_{n-2} >\iff >\forall n\in \Z^+ : \alpha_{1}r_{1}^n+\alpha_{2}+\alpha_{2}r_{2}^n > >\end{align} >$ >Where $\alpha_{1}, \alpha_{2}$ are constants.