To solve the recurrence relation, we first need to find the characteristic equation:
r^2 - 4r - 4 = 0
Using the quadratic formula, we get:
r = (4 ± √32) / 2
r = 2 ± √8
So the general solution is:
an = A(2 + √8)^n + B(2 - √8)^n
Using the initial conditions a0 = 1 and a1 = -1, we can solve for A and B:
a0 = A + B = 1
a1 = A(2 + √8) + B(2 - √8) = -1
Solving for A and B, we get:
A = (-1 - √8) / -4
B = (1 + √8) / 4
Therefore, the solution to the recurrence relation is:
an = [(-1 - √8) / -4](2 + √8)^n + [(1 + √8) / 4](2 - √8)^n
I hope this helps!