Consider the sequence $h_n = (-1)^ng_n$ where $g_{n+2} = g_{n+1} + 20g_n$ and $g_0 = 1$, $g_1 = 1$. Find a second order homogeneous linear recurrence with constant coefficients for $h_n$ and prove that your recurrence is correct for all n.
So I found the linear recurrence to be $h_{n+2} = -h_{n+1} + 20h_n$, but I am stumped on how to prove that it satisfies all $n$, I thought an induction proof was needed but the professor said that it's a one-line proof.
One of the ideas that I had was that, we can write any $g_n$ in terms of a linear combination of $g_0$ and $g_1$ but I am not sure if this would help.
Note that $h_n = (-1)^ng_n\implies g_n = (-1)^nh_n$. Substituting this in the given recurrence gives$$(-1)^{n+2}h_{n+2} = (-1)^{n+1}h_{n+1} + 20(-1)^nh_n$$
I'm sure you can take it from here.