Comment by tomfitz

That's a neat visualisation.

n^2 = sum(1..n) - sum(1..n-1) = 2sum(1..n-1) + n

=> n^2 - n = 2sum(1..n-1) => n(n-1) = 2*sum(1..n-1) => sum(1..n-1) = n(n-1)/2

And you can rewrite that as... sum(1..n) = n(n+1)/2