Prove Theorem $3:$
Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $$r^{k}-c_{1}r^{k-1}-\dots – c_{k} = 0$$
has $k$ distinct roots $r_{1},r_{2},\dots r_{k}.$ Then a sequence $\{a_{n}\}$ is a solution of the recurrence relation $$a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k}$$ if and only if
$$a_{n} = \alpha_{1}r^{n}_{1} + \alpha_{2}r_{2}^{n} + \dots + \alpha_{k}r^{n}_{k}$$
for $n = 0,1,2,\dots,$ where $\alpha_{1},\alpha_{2},\dots,\alpha_{k}$ are constants.