a n+1 ​ −a n ​ =3n characteristic equation: 1/x-1/x^2=01/x−1/x 2 =0 x-1=0x−1=0 x=1x=1 homogeneous solution: a_h=cx^n=ca h ​ =cx n =c particular solution: a_t=An^2+Bn+Ca t ​ =An 2 +Bn+C A(n+1)^2+B(n+1)+C-An^2-Bn-C=3nA(n+1) 2 +B(n+1)+C−An 2 −Bn−C=3n 2An+A+B=3n2An+A+B=3n A=1.5,B=-1.5A=1.5,B=−1.5 a_t=1.5n^2-1.5na t ​ =1.5n 2 −1.5n a_n=a_h+a_t=c+1.5n^2-1.5na n ​ =a h ​ +a t ​ =c+1.5n 2 −1.5n a_0=c+1.5(-1)^2-1.5(-1)=1a 0 ​ =c+1.5(−1) 2 −1.5(−1)=1 c=-2c=−2 a_n=-2+1.5n^2-1.5na n ​ =−2+1.5n 2 −1.5n