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Arithmetic Series Perfect Square

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

    I need to find all arithmetic sequences of integers with the property that the sum of the first n terms is a perfect square for all integers n.

    2. Relevant equations

    a_n = nth term of the sequence = a_1 + (n-1)d
    d = common difference
    Sum of the first n terms of the sequence = n[2a_1+(n-1)d]/2

    3. The attempt at a solution

    I know that the sequence 1, 3, 5, 7, 9... sums to a perfect square every time, as will 1x^2, 3x^2, 5x^2..., with x being any integer.

    This is the only way I can find to make the sequence sum to a perfect square every time. If this isn't the only way, what are the others? If this is, how can I prove it is the only way?
     
  2. jcsd
  3. Feb 26, 2008 #2
    well you noticed that [itex]\sum^n_{m=0} 2m+1 = (n+1)^2[/itex]. Note that [itex](n+1)^2-n^2=2n+1[/itex]. You have made a sequence where adding a successive term makes the sum equal the successive square. What is the [itex](n+2)^2-n^2[/itex]? Certainly you can set a_1 to be a perfect square, then you are guaranteed to have such a sequence that you want. I hope I made enough sense without giving away too much away.
     
    Last edited: Feb 26, 2008
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