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Possible Duplicate:
Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Is there a picture proof for $\sum_{i=1}^{n} i = \frac{n}{2}(n+1)$?

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    Draw an $n$ by $n+1$ rectangular array of lattice points, and split it into two equal halves along an almost diagonal. Taking $n=4$ is probably good enough. Or else equivalently take an $n+1$ times $n+1$ array, erase the main NW to SE diagonal, and slide the lower remaining half up by $1$.2012-03-05

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Draw an $n$ by $n+1$ rectangular array of lattice points, and split it into two equal halves along an almost diagonal. Taking $n=4$ or $n=5$ is probably good enough.

Or else, equivalently, take an $n+1$ by $n+1$ square array of dots, and erase the main Northwest to Southeast diagonal. Then slide the points of the lower remaining half up by $1$.

Remark: Logically speaking, this cannot be an acceptable answer! A request for a proof without words has been answered by using $\dots$ words.

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    @user26345: Your question was very well formulated. I was just making a joke about my answer.2012-03-06