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To be frank, I didn't learn any sort of proof for this (visual or non-visual), so I came up with this proof through trial and error.
Moreover, I haven't checked my proof online yet, therefore I am not sure if I am the first one to come up with this proof - Nonetheless, it is still quite a remarkable proof, at least for me :D.

Hope you will appreciate my visual proof from below!

enter image description here

enter image description here

Shaun
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  • Not bad, but it could be simpler https://math.stackexchange.com/questions/50485/sum-of-n-consecutive-numbers/50514#50514 https://math.stackexchange.com/a/34400/312 – leonbloy Dec 25 '18 at 14:16
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    I'm not sure that offering alternative proofs actually counts as commenting on the poster's one? (Just pointing that out before everyone, myself included, jumps in with their favourite proof and ignores the question!) – timtfj Dec 25 '18 at 16:39

1 Answers1

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Not quite visual, but won't this be simpler?

Write:$$S=1+2+3+\dots +(n-1)+n$$ Reciprocate the order of terms: $$S=n+(n-1)+\dots +3+2+1$$ Add both: $$2S=\underbrace{(n+1)+(n+1)+\dots +(n+1)}_{n \text{ times}}$$ $$2S=n\cdot(n+1)$$

$$S=\frac{n\cdot(n+1)}2$$

Shaun
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pooja somani
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    And this is equivalent to the visual proof where an $n×(n+1)$ rectangle is broken into two "jagged triangles" in like fashion. – timtfj Dec 25 '18 at 16:33
  • @timtfj True! Didn't strike me then... :-/ – pooja somani Dec 25 '18 at 17:16
  • It's Christmas day. One's allowed not to notice things ;-) I think it struck me as a result of looking at the questioner's diagram and seeing a zigzag line. – timtfj Dec 25 '18 at 18:33