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Non-recursive formula

  1. Mar 24, 2005 #1
    How to get a non-recursive formula for y(n):

    y(n)=1 (n=1 or 2)
    y(n)=0 (n=3 or 4)
    y(n)=(y(n-4) + y(n-3))/2


    Any hints apprecaited..
     
  2. jcsd
  3. Mar 24, 2005 #2

    matt grime

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    write down the first few terms, guess an answer and prove it inductively, that'd be my guess.

    or work backwards from y(n) repeatedly subs'ing in and see what works.
     
  4. Mar 24, 2005 #3
    I have already wrote down previous 20 items, still can't find the relationship

    I have already wrote down previous 20 items, still can't find the relationship


     
  5. Mar 24, 2005 #4

    matt grime

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    Ok, I suppose a 4th order is going a little too far to ask you to spot it...

    however, it is linear and homogeneous, so the general solution is of the form At^n for some constants t and A, t satisfies

    t^n = t^{n-4}+t^{n-3}

    or

    t^4=1+t,

    solve that, to get 4 solutions, and then apply the 4 initial conditions.
     
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