Alright, I pretty much have the proof done, now just trying to do the algebra on it. This is the question...
The information I have is:
$$a_k = C_1 r^k + C_2 s^k$$
$$a_{k-1} = C_1 r^{k-1} + C_2 s^{k-1}$$
$$C_1 = \dfrac{s a_0 - a_1}{s - r}$$
$$C_1 = \dfrac{a_1 - r a_0}{s - r}$$
Also have the equation $x^2 = bx + c$ that has two distinct solutions $x = r$ and $x = s$.
What I want / started is ${{a_k}_+}{_1}$, which the formula for that is
$$a_{k+1} = b a_k + c a_{k-1} = b (C_1 r^k + C_2 s^k) + c (C_1 r^{k-1} + C_2 s^{k-1})$$
I know I need to use the fact the $x = s$ and $x = r$ from $x^2 = bx + c$
Hint: Since $r$ and $s$ are the roots of $x^2-bx-c=0$, we have $b=r+s$ and $c=-rs$.
Substitute for $b$ and $c$ in your last displayed equation, and simplify.
As a start, we have $bC_1r^{k}=C_1(r+s)r^k=C_1r^{k+1}+C_1r^ks$. The first term is nice, exactly what we want. And one hopes that the second term will get cancelled (and it will).