Given,
⟹an+1−an=3n
characteristic equation:
1/x−1/x2=0
x−1=0
x=1
Homogeneous solution:
ah=c⋅xn=c⋅(1)n=c
Particular solution:
at=An2+Bn+C
⟹A(n+1)2+B(n+1)+C−An2−Bn−C=3n
⟹2An+A+B=3n
⟹A=1.5, B=−1.5
Hence,
at=1.5n2−1.5n
an=ah+at=c+1.5n2−1.5n
a0=c+1.5(−1)2−1.5(−1)=1
c=−2
an=−2+1.5n2−1.5n
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