a)
The characteristic polynomial of this recurrence relation is
r3−6r2+11r−6=0
r2(r−1)−5r(r−1)+6(r−1)=0
(r−1)(r2−5r+6)=0
(r−1)(r−2)(r−3)=0
r1=1,r2=2,r3=3 Hence, the solutions to this recurrence relation are of the form
an=α1⋅1n+α2⋅2n+α3⋅3n Use the initial conditions
a0=4=α1+α2+α3a1=6=α1+2α2+3α3a2=12=α1+4α2+9α3
α1+α2+α3=4α2+2α3=22α2+6α3=6
α1+α2+α3=4α2+2α3=2α2+3α3=3
α1=3α2=0α3=1Hence, the unique solution to this recurrence relation and the given initial conditions is the sequence {an} with
an=3+3n
b)
The characteristic polynomial of this recurrence relation is
r3−r2−9r+9=0
r2(r−1)−9(r−1)=0
(r−1)(r−3)(r+3)=0
r1=1,r2=−3,r3=3 Hence, the solutions to this recurrence relation are of the form
an=α1⋅1n+α2⋅(−3)n+α3⋅3n Use the initial conditions
a0=6=α1+α2+α3a1=0=α1−3α2+3α3a2=30=α1+9α2+9α3
α1+α2+α3=64α2−2α3=68α2+8α3=24
α1+α2+α3=62α2−α3=3α2+α3=3
α1=3α2=2α3=1Hence, the unique solution to this recurrence relation and the given initial conditions is the sequence {an} with
an=3+2⋅(−3)n+3n
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