How would you solve for the closed term form of $a(n)$ given the general form of the third order linear homogenous recurrence relation with real constant coefficients.
with the initial terms of a1, a2, and a3
and given that the roots of the characteristic equations have
- two repeated roots and a real root
- three repeated roots
(can you give answers for both cases please)
For second order recurrence relations I know that you can use generating functions to deduce a closed form because it is then expressed as a arithmetic series which can be converted into a closed form.
However in the case of the general term of the third order recurrence relations if I follow the same steps what I did with the second order recurrence relation, instead of getting a simple arithmetic series I seemed to get a second order recurrence relation inside the series.
What am I doing wrong?
or is there a different method of approach in this case?
When I search the web I get these results
S(n) = nAx1^n + Bx1^n + Cx2^n,for the case when there are two repeated roots
S(n) = n^2Ax^n + nBx^n + Cx^n, for the case when there are three repeated roots
I just don't know how to get to these results