1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solving a recurrence relation

  1. Mar 23, 2010 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    Solve the recurrence relation
    an = 5an−1 − 3an−2 − 9an−3 for n ≥ 3
    with initial values a0 = 0, a1 = 11, and a2 = 34.

    2. Relevant equations

    its given lol

    3. The attempt at a solution

    I found that the characteristic equation for this rr is x3 - 5x2 + 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

    an = r3n + sn3n - t
    and so plugging back into the give rr we have

    r3n + sn3n - t = 5(r3n-1 + s(n-1)3n-1 - t) - 3(r3n-2 + s(n-2)3n-2 - t) - 9(r3n-3 + s(n-3)3n-3 - 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.
    Last edited: Mar 23, 2010
  2. jcsd
  3. Mar 23, 2010 #2
    Given this, then r,s and t must be equal to what for you to have a0 = 0, a1 = 11, and a2 = 34? This above expression must be valid for all n, after all, not only for [itex]n\geq 3[/itex].

    By the way, you claim that one of your roots is -1; are you sure that the above is entirely correct?
  4. Mar 23, 2010 #3


    User Avatar
    Gold Member

    To see if a root exists, we would plug it into the characteristic equation. When -1 is plugged into the equation, we obtain 0, therefore it is a root of the equation. Corollary, I saw that 3 was a root in same fashion, and found that it was a root of multiplicity 2 when I plugged it into the derivative. This was how we were showed to find the roots.
  5. Mar 23, 2010 #4
    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 3n in the solution, then what would be the term corresponding to -1?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook