Question

Find the generating function of recurrence relation a_(n+1) - a_n = 3n ,n<0 where ao=1

Answer 1

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

 characteristic equation:

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

$x-1=0$

$x=1$

homogeneous solution:

$a_h=cx^n=c$

particular solution:

$a_t=An^2+Bn+C$

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

$2An+A+B=3n$

$A=1.5,B=-1.5$

$a_t=1.5n^2-1.5n$

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

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

$c=-2$

$a_n=-2+1.5n^2-1.5n$