# Arithmetic Series Perfect Square

1. Feb 25, 2008

### Frillth

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. Feb 26, 2008

### jhicks

well you noticed that $\sum^n_{m=0} 2m+1 = (n+1)^2$. Note that $(n+1)^2-n^2=2n+1$. You have made a sequence where adding a successive term makes the sum equal the successive square. What is the $(n+2)^2-n^2$? 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