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Method of Differences problem

  1. Jul 23, 2008 #1
    Hi, I have this question that suppose to use Method of Differences to solve it.

    By using the method of differences, find the sum of the first n terms of the series whose rth term, ur, are

    Ur = (2r - 1)/r(r+1)(r+2)

    I used partial fraction and found.

    Ur = -1/(2r) + 3/(r+1) - 5/[2(r+2)]

    then i did

    -(1/2)[5/(r+2) - 6/(r+1)] - 1/(2r)
    -(1/2)[5/(n+2) - 5/2] - (1/2)(1/r)
    5/4 - 5/(2n+4) - (1/2)[1/(1/2)(n)(n+1)]
    (10n + 20 - 20)/4(2n+4) - 1/(n)(n+1)

    But the answer is wrong. Any help?
  2. jcsd
  3. Jul 23, 2008 #2
    You have made wrong partial fractions. thats not the way.

    first seperate the terms in numerator so Ur = 2/(r+1)(r+2) - 1/r(r+1)(r+2)

    and now splitting them gives

    Ur = 2(1/(r+1) - 1/(R+2)) - 1/2 (1/r(r+1) - 1/(r+1)(r+2)

    so the final Answer will be = 2(1/2 - 1/(r+2)) - 1/2 (1/2 - 1/(r+1)(r+2))
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