The characteristic polynomial of this recurrence relation is
2r3−r2−8r+4=0
(r−2)(2r2+3r−2)=0
(r−2)(r+2)(2r−1)=0The characteristic roots are r1=−2,r2=1/2,r3=2.
Hence, the solutions to this recurrence relation are of the form
an=α1(−2)n+α2(21)n+α3(2)nTo find the constants α1,α2,α3, use the initial conditions. This gives
a0=α1+α2+α3=3
a1=−2α1+21α2+2α3=2
a2=4α1+41α2+4α3=12
α1=1,α2=0,α3=2Hence, the unique solution to this recurrence relation and the given initial
conditions is the sequence {an} with
an=(−2)n+2(2)n (i)
a99=(−2)99+2(2)99=(2)99
(ii)
2a2n+1−a2n=2(−2)2n+1+4(2)2n+1
−((−2)2n+2(2)2n)
=−4(2)2n+8(2)2n−(2)2n−2(2)2n
=(2)2n=(4)n
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