Talk:Closed Form for Triangular Numbers

I've put the proof together for Telescoping Series but I reckon Telescoping Sum would be a stronger result (in that it has wider direct applications and the result for series follows from it) and if we felt real ambitious we could prove it for rings. --Matt Westwood 21:32, 24 December 2008 (UTC)

I put up the recursive proof for triangular numbers. I couldn't find it anywhere else on the internet so I'm not really sure if it's a quality proof, but I put it up anyway :P. 20:19, 24 May 2010 (UTC)

I can't follow why:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S(n)\)	\(=\)	\(\displaystyle n + (n-1) + (n-2) + \cdots + 2 + 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle n + (n-1) + (n-2) + \cdots + (n-(n-2)) + (n-(n-1))\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle n^2 - (1 + 2 + \cdots + (n-1))\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

... can you elaborate? --Matt Westwood 20:47, 24 May 2010 (UTC)

Well... the first eqn you stated should be obvious right? For example, $S(5) = 5 + 4 + 3 + 2 + 1$. For the second eqn, I'm just restating the constants in terms of $n$. $n-(n-2) = 2$ and $n-(n-1) = 1$; there's no arguing that. For the third eqn I'm just using the associative/commutative rule for addition/subtraction. The right side of the second eqn can be intermediately expressed as:

\(\displaystyle \)

\(\displaystyle \)

\(\displaystyle \)

\(\displaystyle \)

\(=\)

\(\displaystyle (n_1 + n_2 + \cdots + n_n) - 1 - 2 - \cdots - (n-1)\)

\(\displaystyle \)

\(\displaystyle \)

\(\displaystyle \)

And further simplifying that results in the third eqn.

Oh yes of course. Thanks. --Matt Westwood 22:12, 25 May 2010 (UTC)

Landmark

The point has been made that this was the first proof. However, technically speaking, that was held by Closed Form for Triangular Numbers/Direct Proof (which already has the landmark flag on it). The other proofs came later. --prime mover 16:52, 14 March 2012 (EDT)

"This article is a landmark page. It was the 1st theorem on ProofWiki!"? --GFauxPas 17:28, 14 March 2012 (EDT)

Whatever. I'm starting to suffer from one of my periodical bouts of inability to donate a damn about this site at the moment. Sorry. --prime mover 17:51, 14 March 2012 (EDT)

A speedy recovery. --GFauxPas 18:12, 14 March 2012 (EDT)

Good night's sleep is what I needed. Your idea is sound. --prime mover 04:15, 15 March 2012 (EDT)

Re: Khan Academy link, http://code.google.com/p/khanacademy/issues/detail?id=9790 --GFauxPas 08:42, 15 March 2012 (EDT)

This is the main reason why I'm not keen on citing other websites from here - because the nature of the web makes the links too unreliable. On Khan Academy it happens far too often. --prime mover 11:07, 15 March 2012 (EDT)