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Solving a recurrence relation 
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#1
Mar2310, 10:39 AM

PF Gold
P: 117

1. The problem statement, all variables and given/known data
Solve the recurrence relation a_{n} = 5a_{n−1} − 3a_{n−2} − 9a_{n−3} for n ≥ 3 with initial values a_{0} = 0, a_{1} = 11, and a_{2} = 34. 2. Relevant equations its given lol 3. The attempt at a solution I found that the characteristic equation for this rr is x^{3}  5x^{2} + 3x + 9 and found that the characteristic roots are 3, 3, 1...because we have 2 indistinct roots, I multiplied one of the 3 terms by n to get a_{n} = r3^{n} + sn3^{n}  t and so plugging back into the give rr we have r3^{n} + sn3^{n}  t = 5(r3^{n1} + s(n1)3^{n1}  t)  3(r3^{n2} + s(n2)3^{n2}  t)  9(r3^{n3} + s(n3)3^{n3}  t) I'm thinking that in order to solve this, we're going to have to set this up as a system of equations, but I'm not sure how to do that with what I have. Any hints/tips/ suggestions on where to go next would be very helpful. 


#2
Mar2310, 01:07 PM

P: 402

By the way, you claim that one of your roots is 1; are you sure that the above is entirely correct? 


#3
Mar2310, 08:23 PM

PF Gold
P: 117




#4
Mar2310, 08:28 PM

P: 402

Solving a recurrence relation
I don't have any doubt that 1 is a root: it is. But, if 3 is a root (forget the multiplicity for a moment) and it gives rise to a term 3^{n} in the solution, then what would be the term corresponding to 1?



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