Arithmetic Series Perfect Square

[Frillth]

Homework Statement

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.

Homework 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

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?

 

[jhicks]

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.