Question

a_n=3a_n-1+(n²+n-2)3ⁿ

Answer 1

$a_n=3a_{n-1}+\left( n^2+n-2 \right) 3^n\\Homogeneous\,\,equation:\\a_n=3a_{n-1}\Rightarrow a_n=C\cdot 3^n\\Let\,\,a_n=C\left( n \right) \cdot 3^n\\C\left( n \right) \cdot 3^n=3\cdot C\left( n-1 \right) \cdot 3^{n-1}+\left( n^2+n-2 \right) 3^n\\C\left( n \right) =C\left( n-1 \right) +n^2+n-2=\\=C\left( n-2 \right) +\left( n-1 \right) ^2+\left( n-1 \right) -2+n^2+n-2=...=\\=C\left( 0 \right) +1^2+2^2+...+n^2+1+2+...+n-2-2-...-2=\\=C\left( 0 \right) +\frac{n\left( n+1 \right) \left( 2n+1 \right)}{6}+\frac{n\left( n+1 \right)}{2}-2n=\\=C\left( 0 \right) +\frac{n^3+3n^2-4n}{3}\\a_n=\left( C\left( 0 \right) +\frac{n^3+3n^2-4n}{3} \right) \cdot 3^n=\left( n^3+3n^2-4n+C \right) \cdot 3^{n-1}$