I am currently struggling to prove the following:
Let $\left(D_{n}\right)_{n=1}^{+\infty}$ be a complex sequence that follows: $D_{n}=p D_{n-1}+qD_{n-2}$ for every $n\geq 3$ where $p, q \in \mathbb{C}$ and $p \neq 0 \vee q \neq 0$. For $q \neq 0$ let $\lambda_{1}, \lambda_{2}$ be the solutions of $x^{2}=p x+q$
i. for $\lambda_{1} \neq \lambda_{2}$ show that: $D_{n}=c_{1} \lambda_{1}^{n}+c_{2} \lambda_{2}^{n}$ for $n \geq 1,$ where $c_{1}, c_{2}$ are constants determined by $D_{1}$ a $D_{2}$
ii. for $\lambda_{1}=\lambda_{2}$ show that : $D_{n}=\left(c_{1}+n c_{2}\right) \lambda_{1}^{n}$ for $n \geq 1,$ where $c_{1}, c_{2}$ are constants determined by $D_{1}$ a $D_{2}$.
I dont really know where to start with the proof or even how to understand the statement. I know that this sequence is later used to compute the determinant of tridiagonal matrix; however, I am not completly sure that this information is of any use.