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Proof for formula for sum of sequence 1+2+3+…+n?

Proof without words:

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad $ enter image description here

How does this image prove the identity $1+2+3+4\cdots + (n-1) = \binom{n}{2}$?

I found this here; could anybody explain this in a lucid manner?

Community
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Quixotic
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1 Answers1

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This shows that every yellow circle uniquely determines a pair of blue circles and vice versa. The number of yellow ones is the LHS, the number of pairs of blue ones is the RHS. Cute!

Max
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  • I don't think I can see the proof yet :/ – Quixotic Oct 28 '11 at 23:37
  • There are 3 steps (ok maybe 4). 1)every yellow circle uniquely determines a pair of blue circles and vice versa 2)The number of yellow ones is the LHS 3)the number of pairs of blue ones is the RHS. 4)All of this implies the equality we want. Which of these are confusing? – Max Oct 28 '11 at 23:39
  • was confusing,but I guess I got it,is it like going in same the direction as shown in the example for every other yellow discs?
    • – Quixotic Oct 28 '11 at 23:41
  • and number 4 too. – Quixotic Oct 28 '11 at 23:42
  • Exactly. And conversely, from any pair of blue discs you can go up in these directions and the lines will intersect at a yellow disc. – Max Oct 28 '11 at 23:44
  • You should be able to figure out 4 (as in: 1)y=pb 2)y=lhs 3)pb=rhs imply 4)rhs=lhs). – Max Oct 28 '11 at 23:46