Solve the inhomogeneous recurrence relation
a_{n}=3a_{n-1}+n^{2}-3 with a_{0}=1
Any help please? I tried a few different cases which got me nowhere.
You may use the process of symbolic differencing to get a homogeneous recurrence. We are given:
(1)
Now, replace with to get:
(2)
Now, subtract (1) from (2) to get another recurrence. Repeat this process until you have a homogeneous recurrence, from which you may determine the closed form from the characteristic roots, and then determine the parameters from the initial values.
Think of recurrence equations as being like a discrete type of differential equation.
1) Solve the homogeneous case: to get
2) Now use variation of parameters. Try:
You can further simply that.