# For set $X$ of integers, why is the square of the sum of its elements equal to the sum of pairwise products?

title pretty much says it all:

$\sum_{i \in X} \sum_{j \in X} ij = (\sum_{i \in X}i)^2$

I'm trying to find out why two ways of writing the same formula are identical, and this is what it comes down to. I find this to be true for all cases I look at (and I assume it is, because the equality of the two original formulations is pretty well-established), but I have no idea why, nor how to proceed such a kind of question.

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This can be demonstrated very neatly by a graphical argument: Draw a square with sides $\sum_{i\in X}i$, and calculate the area in 2 different ways. – John Wordsworth Jun 14 '12 at 16:55
thanks @OldJohn, beautiful visualization! – Nicolas Jun 14 '12 at 17:26