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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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Lori Jiang
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    http://www.wolaver.org/teaching/clip_image002.gif – Joel Cohen Mar 05 '12 at 23:32
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    See this answer: http://math.stackexchange.com/questions/2260/proof-for-formula-for-sum-of-sequence-123-ldotsn/2288#2288 – Aryabhata Mar 05 '12 at 23:40
  • Even though the questions are different, I have voted to close a dupe. Any proof without words can be added to that question. – Aryabhata Mar 05 '12 at 23:42
  • 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 $1$. – André Nicolas Mar 05 '12 at 23:45

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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.

André Nicolas
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