Solve the recurrence relation
an = 5an−1 − 3an−2 − 9an−3 for n ≥ 3
with initial values a0 = 0, a1 = 11, and a2 = 34.
its given lol
The Attempt at a Solution
I found that the characteristic equation for this rr is x3 - 5x2 + 3x + 9 and found that the characteristic roots are 3, 3, -1...because we have 2 indistinct roots, I multiplied one of the 3 terms by n to get
an = r3n + sn3n - t
and so plugging back into the give rr we have
r3n + sn3n - t = 5(r3n-1 + s(n-1)3n-1 - t) - 3(r3n-2 + s(n-2)3n-2 - t) - 9(r3n-3 + s(n-3)3n-3 - t)
I'm thinking that in order to solve this, we're going to have to set this up as a system of equations, but I'm not sure how to do that with what I have. Any hints/tips/ suggestions on where to go next would be very helpful.