My visual interpretation of $1+2+3+ dots +n$












12












$begingroup$


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






summation visualization







share|cite|improve this question











$endgroup$












  • $begingroup$
    Not bad, but it could be simpler math.stackexchange.com/questions/50485/… math.stackexchange.com/a/34400/312
    $endgroup$
    – leonbloy
    Dec 25 '18 at 14:16








  • 1




    $begingroup$
    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!)
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:39
















12












$begingroup$


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






summation visualization







share|cite|improve this question











$endgroup$












  • $begingroup$
    Not bad, but it could be simpler math.stackexchange.com/questions/50485/… math.stackexchange.com/a/34400/312
    $endgroup$
    – leonbloy
    Dec 25 '18 at 14:16








  • 1




    $begingroup$
    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!)
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:39














12












12








12


2



$begingroup$


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






summation visualization







share|cite|improve this question











$endgroup$




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






summation visualization




summation visualization






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 25 '18 at 14:35









Shaun

9,083113683




9,083113683










asked Dec 25 '18 at 14:05







user629248



















  • $begingroup$
    Not bad, but it could be simpler math.stackexchange.com/questions/50485/… math.stackexchange.com/a/34400/312
    $endgroup$
    – leonbloy
    Dec 25 '18 at 14:16








  • 1




    $begingroup$
    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!)
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:39


















  • $begingroup$
    Not bad, but it could be simpler math.stackexchange.com/questions/50485/… math.stackexchange.com/a/34400/312
    $endgroup$
    – leonbloy
    Dec 25 '18 at 14:16








  • 1




    $begingroup$
    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!)
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:39
















$begingroup$
Not bad, but it could be simpler math.stackexchange.com/questions/50485/… math.stackexchange.com/a/34400/312
$endgroup$
– leonbloy
Dec 25 '18 at 14:16






$begingroup$
Not bad, but it could be simpler math.stackexchange.com/questions/50485/… math.stackexchange.com/a/34400/312
$endgroup$
– leonbloy
Dec 25 '18 at 14:16






1




1




$begingroup$
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!)
$endgroup$
– timtfj
Dec 25 '18 at 16:39




$begingroup$
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!)
$endgroup$
– timtfj
Dec 25 '18 at 16:39










1 Answer
1






active

oldest

votes


















4












$begingroup$

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: $$2S=underbrace{(n+1)+(n+1)+dots +(n+1)}_{n text{ times}}$$
$$2S=ncdot(n+1)$$




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







share|cite|improve this answer











$endgroup$









  • 2




    $begingroup$
    And this is equivalent to the visual proof where an $n×(n+1)$ rectangle is broken into two "jagged triangles" in like fashion.
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:33










  • $begingroup$
    @timtfj True! Didn't strike me then... :-/
    $endgroup$
    – idea
    Dec 25 '18 at 17:16










  • $begingroup$
    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.
    $endgroup$
    – timtfj
    Dec 25 '18 at 18:33













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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









4












$begingroup$

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: $$2S=underbrace{(n+1)+(n+1)+dots +(n+1)}_{n text{ times}}$$
$$2S=ncdot(n+1)$$




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







share|cite|improve this answer











$endgroup$









  • 2




    $begingroup$
    And this is equivalent to the visual proof where an $n×(n+1)$ rectangle is broken into two "jagged triangles" in like fashion.
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:33










  • $begingroup$
    @timtfj True! Didn't strike me then... :-/
    $endgroup$
    – idea
    Dec 25 '18 at 17:16










  • $begingroup$
    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.
    $endgroup$
    – timtfj
    Dec 25 '18 at 18:33


















4












$begingroup$

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: $$2S=underbrace{(n+1)+(n+1)+dots +(n+1)}_{n text{ times}}$$
$$2S=ncdot(n+1)$$




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







share|cite|improve this answer











$endgroup$









  • 2




    $begingroup$
    And this is equivalent to the visual proof where an $n×(n+1)$ rectangle is broken into two "jagged triangles" in like fashion.
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:33










  • $begingroup$
    @timtfj True! Didn't strike me then... :-/
    $endgroup$
    – idea
    Dec 25 '18 at 17:16










  • $begingroup$
    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.
    $endgroup$
    – timtfj
    Dec 25 '18 at 18:33
















4












4








4





$begingroup$

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: $$2S=underbrace{(n+1)+(n+1)+dots +(n+1)}_{n text{ times}}$$
$$2S=ncdot(n+1)$$




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







share|cite|improve this answer











$endgroup$



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: $$2S=underbrace{(n+1)+(n+1)+dots +(n+1)}_{n text{ times}}$$
$$2S=ncdot(n+1)$$




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








share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 25 '18 at 14:34









Shaun

9,083113683




9,083113683










answered Dec 25 '18 at 14:12









ideaidea

2,15841125




2,15841125








  • 2




    $begingroup$
    And this is equivalent to the visual proof where an $n×(n+1)$ rectangle is broken into two "jagged triangles" in like fashion.
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:33










  • $begingroup$
    @timtfj True! Didn't strike me then... :-/
    $endgroup$
    – idea
    Dec 25 '18 at 17:16










  • $begingroup$
    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.
    $endgroup$
    – timtfj
    Dec 25 '18 at 18:33
















  • 2




    $begingroup$
    And this is equivalent to the visual proof where an $n×(n+1)$ rectangle is broken into two "jagged triangles" in like fashion.
    $endgroup$
    – timtfj
    Dec 25 '18 at 16:33










  • $begingroup$
    @timtfj True! Didn't strike me then... :-/
    $endgroup$
    – idea
    Dec 25 '18 at 17:16










  • $begingroup$
    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.
    $endgroup$
    – timtfj
    Dec 25 '18 at 18:33










2




2




$begingroup$
And this is equivalent to the visual proof where an $n×(n+1)$ rectangle is broken into two "jagged triangles" in like fashion.
$endgroup$
– timtfj
Dec 25 '18 at 16:33




$begingroup$
And this is equivalent to the visual proof where an $n×(n+1)$ rectangle is broken into two "jagged triangles" in like fashion.
$endgroup$
– timtfj
Dec 25 '18 at 16:33












$begingroup$
@timtfj True! Didn't strike me then... :-/
$endgroup$
– idea
Dec 25 '18 at 17:16




$begingroup$
@timtfj True! Didn't strike me then... :-/
$endgroup$
– idea
Dec 25 '18 at 17:16












$begingroup$
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.
$endgroup$
– timtfj
Dec 25 '18 at 18:33






$begingroup$
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.
$endgroup$
– timtfj
Dec 25 '18 at 18:33




















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