- #1

Frillth

- 80

- 0

## 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?