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Homework Help: How to find a difference polynomial?

  1. Nov 25, 2008 #1
    1. The problem statement, all variables and given/known data

    Let {a_n} be the sequence 3, 4, 8, 17,..., where a_0 = 3 and a_n+1 = a_n + (n+1)^2, n greater than or equal to 0. Find a polynomial such that a_n = f(n)

    2. Relevant equations

    f(n) = summation from r=0 to infinity of C(n,r) delta^r a_0

    3. The attempt at a solution

    I just have no idea how to start it. I have: f(n) = 3C(n, 0) + 4C(n, 1) + 8C(n,2) + something?
     
  2. jcsd
  3. Nov 26, 2008 #2

    Office_Shredder

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    Look at the differences
    4-3=1
    8-4=4
    17-8=9

    1,4,9,... is just the sequence for n2

    So f(n+1)-f(n) = n2. If the difference has degree k, the actual polynomial has degree k+1 (something you should try proving) so find a cubic polynomial with this characteristic
     
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