There is a recurrence relation such that

$$T(0) = 0, T(1) = 1$$

$$T(n) = 7T(n-1) -12T(n-2)$$

And I'm trying to prove that $$T(n) = -3^n + 4^n$$

So far I have the following

Check base cases for 0,1

$$T(0) = -3^0 + 4^0 = 0$$
$$T(1) = -3^1 + 4^1 = 1$$

As both of these are true we can continue

Next we assume that (inductive hypothesis) $$T(n) = -3^n + 4^n$$

And for the inductive step we have $$T(n+1) = -3^{n+1} + 4^{n+1}$$

We know from our recurrence relation that $$T(n + 1) = 7T(n) - 12T(n-1)$$

This is where I am stuck. I know I can substitute in T(n) but I'm not sure what to do for T(n-1)