# How to prove $\sum_{i=1}^{n} i = \frac{n}{2}(n+1)$ without words? [duplicate]

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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## marked as duplicate by Aryabhata, Sivaram Ambikasaran, Asaf Karagila, Listing, Kannappan Sampath Mar 6 '12 at 3:52

See this answer: math.stackexchange.com/questions/2260/… –  Aryabhata Mar 5 '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 5 '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 5 '12 at 23:45
Thank you all. @AndréNicolas I will accept this as an answer. –  Lori Jiang Mar 5 '12 at 23:51

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.