1

Possible Duplicate:
Proof for formula for sum of sequence $1+2+3+\ldots+n$?

I have a question:

Show that $$1+2 + \cdots+ n = \binom{n+1}{2}$$

This is a combinatorial proof. Here is my thinking:

Proof. ($\Leftarrow$): Suppose we have $n+1$ objects labeled $1$ to $n+1$. Then there are $n$ ways to pair $1$ with the other objects, $n-1$ ways to pair $2$ with the other objects (since $12$ is the same as $21$), etc.

Is that correct? How would we do the other direction?

user1176409
  • 1
lary
  • 21
  • Your argument is not complete, but that is the correct idea. Cf: http://math.stackexchange.com/questions/2260/proof-for-formula-for-sum-of-sequence-123-ldotsn/ What other direction? – Daniel Pietrobon Nov 11 '12 at 10:53
  • That question doesn't ask for a combinatorial proof, but there are enough answers that provide one, so I also voted to close as duplicate of that. – joriki Nov 11 '12 at 11:40
  • About "the other direction": There's an "other direction" in proving equivalences, because an equivalence $\Leftrightarrow$ can be considered as the conjunction of two statements, $\Rightarrow$ and $\Leftarrow$. The analogue for an equality would be to consider it as a conjunction of two statements, $\ge$ and $\le$; then there would be two directions. However, that's not what you did, you proved (or started to prove) the equality itself, and when you do that, there's no other direction left to prove. – joriki Nov 11 '12 at 11:43

0 Answers0