Answer to Question #216929 in Discrete Mathematics for Kiran

Question #216929

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


1
Expert's answer
2021-10-29T02:52:39-0400

Given,

    an+1an=3n\implies a_{n+1}-a_n=3n


characteristic equation:

1/x1/x2=01/x-1/x^2=0

x1=0x-1=0

x=1x=1


Homogeneous solution:

ah=cxn=c(1)n=ca_h=c\cdot x^n=c\cdot (1)^n=c


Particular solution:

at=An2+Bn+Ca_t=An^2+Bn+C

    A(n+1)2+B(n+1)+CAn2BnC=3n\implies A(n+1)^2+B(n+1)+C-An^2-Bn-C=3n

    2An+A+B=3n\implies 2An+A+B=3n

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

Hence,

at=1.5n21.5na_t=1.5n^2-1.5n


an=ah+at=c+1.5n21.5na_n=a_h+a_t=c+1.5n^2-1.5n

a0=c+1.5(1)21.5(1)=1a_0=c+1.5(-1)^2-1.5(-1)=1

c=2c=-2


an=2+1.5n21.5n\boxed{a_n=-2+1.5n^2-1.5n}

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