1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    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!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook