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Ishaan94

  • 2 years ago

Whom of you can explain it to me? http://jeremykun.files.wordpress.com/2011/06/triangle-proof.png

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  1. Ishaan94
    • 2 years ago
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    http://mathoverflow.net/questions/8846/proofs-without-words/69756#69756

  2. Ishaan94
    • 2 years ago
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    If it's fairly easy and you think I am not pushing myself please do tell me, instead of giving out the answer.

  3. asnaseer
    • 2 years ago
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    BTW: Nice site.

  4. asnaseer
    • 2 years ago
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    @Ishaan94 - do you know how to get the sum of all the components in the last triangle?

  5. Ishaan94
    • 2 years ago
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    No ._.

  6. Ishaan94
    • 2 years ago
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    wait

  7. Ishaan94
    • 2 years ago
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    Is it like this? (2n+1)+2(2n+1)+3(2n+1)+...+n(2n+1)

  8. asnaseer
    • 2 years ago
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    exactly - now factor out the (2n+1)

  9. hosiduy
    • 2 years ago
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    i think like this way: i will assume that this is true until line n+1 (go downward from first line), i have: first line is true, 1+n+n = 2n + 1. so with line: n+1 we have: (n+1) + ... (n+1) at first triangle, at second triangle, we have: 1+ 2 + ... + n+ n+1, and at third: n+1 + n + ... + 1, sum last line, we still get: 2(n+1) + 1 + .... 2(n+1) + 1

  10. Ishaan94
    • 2 years ago
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    \[\frac{n\left(n+1\right) \left(2n+1\right)}2\]What about the six?

  11. asnaseer
    • 2 years ago
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    that is correct - now you have to subtract the sum of the previous 2 triangles from this

  12. Ishaan94
    • 2 years ago
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    but why?

  13. asnaseer
    • 2 years ago
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    what you are trying to find is the sum of the numbers in the 1st triangle

  14. asnaseer
    • 2 years ago
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    \[1^1+2^2+3^3+...\]

  15. asnaseer
    • 2 years ago
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    that should have started with \(1^2\)

  16. asnaseer
    • 2 years ago
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    it is a visual proof of this sum

  17. Ishaan94
    • 2 years ago
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    oh I see it

  18. asnaseer
    • 2 years ago
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    It's an interesting proof - I suggest you try and do it yourself - you'll have greater pleasure from that :)

  19. Ishaan94
    • 2 years ago
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    thanks, i will try my best.

  20. asnaseer
    • 2 years ago
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    let me know if you require a clue.

  21. asnaseer
    • 2 years ago
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    when you do "see" how to do it - it is quite remarkable!

  22. asnaseer
    • 2 years ago
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    @Ishaan94 - try turning your head as you view the triangles on the left...

  23. asnaseer
    • 2 years ago
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    this really is a "visual" solution - so don't think about any complex math here

  24. Ishaan94
    • 2 years ago
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    OMG this is the best proof I have seen in my life so far

  25. asnaseer
    • 2 years ago
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    :) glad you "saw" it at last!

  26. asnaseer
    • 2 years ago
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    it is a very pleasing proof

  27. Ishaan94
    • 2 years ago
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    it is. and thanks a lot asnaseer. :-)

  28. asnaseer
    • 2 years ago
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    yw :)

  29. rational
    • 7 months ago
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    really beautiful xD

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