sum of odd numbers

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sum of even numbers

And adding 1 to each number, in Mr. pahio entry (Hi my friend!), we get the sum of n first even numbers, i.e. n^2+n=n(n+1).

Geometric proof

There's also the standard geometric 'proof' of this statement that you might want to add. Draw a square matrix of dots. Then the size of the set of dots
\{S_n=(x,y) | x\leq n, y=n or y\leq n, x=n\}
is 2n+1; regarding each of these as a shell, as you add shells, you clearly get the next square.

Roger

Re: Geometric proof

Roger, you mean the $\lnot$-formed patterns of unit squares which form the n times n square. I think this idea may be a bit too complex for to be added to that entry -- at least I cannot explain it sufficiently clearly. But if you like, feel free and add it.

Jussi