Find the generating function of recurrence relation an+1_an=3n ,n less than 0 where ao=1

Given,

$\implies a_{n+1}-a_n=3n$

characteristic equation:

$1/x-1/x^2=0$

$x-1=0$

$x=1$

Homogeneous solution:

$a_h=c\cdot x^n=c\cdot (1)^n=c$

Particular solution:

$a_t=An^2+Bn+C$

$\implies A(n+1)^2+B(n+1)+C-An^2-Bn-C=3n$

$\implies 2An+A+B=3n$

$\implies A=1.5,\ \ B=-1.5$

Hence,

$a_t=1.5n^2-1.5n$

$a_n=a_h+a_t=c+1.5n^2-1.5n$

$a_0=c+1.5(-1)^2-1.5(-1)=1$

$c=-2$

$\boxed{a_n=-2+1.5n^2-1.5n}$