Explanation of Linear Nonhomogeneous Recurrence Relations Problem

Question

$$5\times 3^n=v_{n+2}-6v_{n+1}+9v_n$$ $$=C(n+2)^23^{n+2}-6C(n+1)^23^{n+1}+9Cn^23^n$$ $$=18C3^n$$

Can anyone explain to me how he got $18C3^n$. I've been simplifying the 2nd step but haven't gotten close to that answer. Thanks!

Answer 1

You have: $9C(n+2)^2 - 18C(n+1)^2 + 9Cn^2 = 9C(n^2+4n+4) - 18C(n^2+2n+1) + 9Cn^2=18C$. Now multiply both sides by $3^n$ to get the answer.