How to solve the following non-homogenous linear recurrence relation

Question

find the particular solution for:

can someone show me how to get the answer for this?

Answer 1

The particular solution will be of the form $a_n=\lambda n+\mu$ as the only function within the recurrence relation used is a polynomial of degree $1$. Plugging in this solution gives $$\lambda n+\mu=\lambda (n-1)+\mu+\lambda (n-2)+\mu+2n$$ $$\lambda n+\mu=2\lambda n+2\mu-3\lambda+2n$$ $$\lambda n+\mu-3\lambda=-2n$$ $$\implies\lambda=-2,\mu=-6$$ Hence the particular solution is $$a_n=-2n-6$$