Tri-Nums Sums! (visual proof IV)

This is a short, animated visual proof showing how to find the sum of the first n triangular numbers (which themselves are sums of the first n integers) using stacks of tokens on three triangular stacks of squares.
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Here are some related videos with sums of triangular numbers:
• 3D Sum of Triangular N...
• Sum of Triangular Numb...
• Sum of Triangular Numb...
• Geometry of the Twelve...
Here is a sum of squares visual proof using a similar technique:
• Sum of Squares II (vis...
This animation is based on a visual proof attributed to Richard K. Guy by Roger B. Nelsen in his first Proofs without Words compendium (I may receive a small commission at no cost to you for these affiliate links):
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[@bmx666bmx666]

I didn't understand the part about Tn = n(n+1)/2, then I remembered that it's just the sum of 1+2+3...+(n-1)+n

[@Humman0id]

3 minutes wth

[@MathVisualProofs]

?

[@alanthayer8797]

ULTRA NICE VISUALS to provide Ultra Clarity! Thanks ova n ova again! Keepm comin I watch all !

[@MathVisualProofs]

Thanks!

[@Learnwithme.167]

Please someone reply 👇👇
Draw two parallel lines and connect them by two non parallel equal line then draw diagonal dividing that shape into two triangles
Will the triangles be congruent ?
those triangles can be proven congruent by SSA axiom
1 side is given equal
1 side is common side
1 angle is equal due to parallel lines we had drawn first (I am not taking about non parallel equal lines)
When the triangles are congruent then their corresponding angles will be equal then those non parallel lines will be parallel because alternate angles are equal in the name of corresponding angles
Am I misunderstanding something?

[@realitant]

I think this one is actually probably quicker to see through generating functions

[@aoifedeborha2420]

It’s a beautiful visual, I’m just wondering how can you remember that it’s specifically in those directions? Does it even make a difference?

[@MathVisualProofs]

Directions doesn’t make a difference. You want to make sure the tokens are laid out in a way so that when you overlay the three there are n+2 on each cell.

[@aoifedeborha2420]

@@MathVisualProofs thank you!!! 😊

[@math_travel]

your solution is different from what I thought. it is what I see for the first time. however it is so interesting. furthermore the image is also best. thanks~

[@MathVisualProofs]

😀👍

[@furnaceheadgames9001]

What about triangle roots

[@mohannad_139]

wait a second, isn't this formula the same as the sum of first n squares? 1² + 2² + 3² + 4² + ... + n²

[@tushar999_]

No , it is n(n+1)(2n+1) /2

[@MathVisualProofs]

Very close. Both are on my channel numerous times and they have similar visual proofs.

[@jakobthomsen1595]

Very nice!

[@MathVisualProofs]

Thanks! 😀