Talk:Triangular number/Archive 1

666
666 is one of the most famous triangular numbers, wherever its fame comes from, and it's known as the "Number of the Beast" first from the Bible, only secondarily from numerology. Many people would have no idea what its numerological significance is, but would know what it meant in the Bible. — Preceding unsigned comment added by Jacquerie27 (talk • contribs) 16:38, 24 April 2003 (UTC)

Ouzo Cross
'''Triangular numbers are presented in "The Ouzo Prophecy," and provide the basis for the construction of the Ouzo Cross.

Robert Merlin Evenson/Church of Ouzo

bobevenson@yahoo.com. — Preceding unsigned comment added by 208.30.83.3 (talk) 22:25, 25 July 2005 (UTC)

I have added some links to a video Podcast and they were remove
Hi everyone I have added some links to a video podcast that I own. I think they are a nice addition to wikipedia please look at them and express you oppinion here , judge for yourself if the links are really useful or not to wikipedia.


 * Video PodCast by http://www.isallaboutmath.com/

If any of you think they are valuable to wikipedia then feel free to add them back in the external links.

Regards SilentVoice 03:22, 22 January 2007 (UTC)


 * If your material is sincerely useful, please consider adding to other wikis that are more inclusive, such as wikinfo and wikiknowledge.--69.87.194.231 23:21, 11 February 2007 (UTC)

Right angle triangle
Triangular numbers may also be graphically represented by a right angle triangle. Percussim 09:15, 25 April 2007 (UTC)

Reason for cleanup?
I don't know why this article was tagged for clean up. Having read it a couple of times, the only thing that to me seemed to need cleaning up was the Maple example, which I decided to remove (right decision, wrong decision, I'm not sure). Any reader who has Maple ought to be able to figure out how to use the formulas given here to instruct Maple to calculate these numbers. And if not, he can always look it up in the OEIS. PrimeFan 23:17, 25 February 2007 (UTC)


 * there are so many theorems and interesting relations between tri#'s and other #'s that it would be difficult to present all of it without the article reading like a long trivia section. i removed that eye-sore table in the introduction.Essap 02:30, 5 May 2007 (UTC)

Floor of square root
the part that said n = the floor of sqrt(2x) is misleading at best. if you put for example x=8, you get n=4, but 8 is not the 4th triangular number. i removed it becuase of that.Essap 02:36, 17 May 2007 (UTC)

Not quite perfect
http://en.wikipedia.org/wiki/Perfect_number should probably be read carefully, 10 isn't a perfect number and yet is a triangular number. — Preceding unsigned comment added by 90.240.83.49 (talk) 22:52, 28 June 2007 (UTC)

Triangle root formula
It is good to see that others have found what I found about 10 years ago: a triangle root formula. Once I discovered it, I saw that I could extract the root of any polygonal number in existence. It's the 3D numbers that now have me stumped. I know how to generate the tetrahedral and octahedral numbers but have yet to find a way to extract their roots. Any help here is welcome. :) FYI: I am the moderator for http://forums.delphiforums.com/figurate/start — Preceding unsigned comment added by R3hall (talk • contribs) 23:28, 30 June 2007 (UTC)

"Whenever a triangular number is divisible by 3, one third of it will be a pentagonal number"
not true, for instance 6, 21 --86.143.232.149 12:26, 10 July 2007 (UTC).

Not quite perfect continued
The claim is made in this article that all even perfect numbers are also triangular, but the second (third) perfect number, 28, isn't triangular, as far as I can figure out. am I missing something?

Samois98 04:15, 10 July 2007 (UTC)


 * 28 is the 6th (or thereabouts... short term memory problems) triangular number.66.216.172.3 16:51, 10 July 2007 (UTC)

"Triangular numbers ... describe numbers of balls that can be arranged in a triangle."
Can someone elaborate on this in the article? It isn't clear to me why I can't fit a non-triangular number of balls into a triangle, or just what sort of balls-into-triangle sense is meant. 66.216.172.3 16:51, 10 July 2007 (UTC)
 * Yes, this should be made clearer here. Take a look at the graphic at Figurate number to see what kind of triangles is meant.  Cheers, Doctormatt 07:52, 14 July 2007 (UTC)
 * That's what I (eventually) assumed it meant. I'm not sure what the a simple way to describe that would be, other than maybe linking to Figurate number.  Does anyone have a good idea?  I'll think about it and see if I come up with anything. 207.103.181.5 16:40, 17 July 2007 (UTC)


 * Games: Here's an image from bowling illustrating $$T_4$$: Ten-pin bowling. And here's one from pool illustrating $$T_5$$: Eight-ball. /84.238.113.244 (talk) 21:58, 16 March 2008 (UTC)

Confusing Wording
"A triangular number is the number of dots in an equilateral triangle evenly filled with dots." What does it mean for the triangle to be evenly filled versus unevenly filled? Why not just "filled"? —Preceding unsigned comment added by 71.193.118.77 (talk) 03:36, 3 March 2009 (UTC)

Formula
I added the simplest formula for finding any given triangle number I could come up with (surprised it wasn't on here) and a little bit of trivia about the formula. I'll be checking back later (this knowledge is the result of a 14 hour plane ride! I want to make sure it's respected) although I may be under a different IP than this one. --24.245.11.103 12:31, 15 May 2007 (UTC)


 * It was on there. In the second line. —David Eppstein 15:40, 15 May 2007 (UTC)


 * Oops, I stopped reading that line after the big E looking thing. Sorry, mate! --134.84.5.63 18:43, 15 May 2007 (UTC)


 * In the initial formula at the top of the page, there seems to be an incorrect deduction, only in the final equation with the reduction of the n squared plus n all over 2, down to (n+1)/2, unless I'm missing a symbol or something (font?). Please check. RS — Preceding unsigned comment added by 151.201.43.241 (talk) 02:44, 2 July 2008 (UTC)


 * In the mathml it is (n+1) "\choose" 2. I have no clue what \choose is, and am disappointed to have this definition not rendered in the article and also not explained.  Is it incorrect or simply impenetrable? 64.127.105.60 (talk) 22:11, 8 June 2009 (UTC)


 * The symbol \choose is a LaTeX symbol that is used to type e.g. binomial coefficient. There is a link in the following text to the article about binomial coefficient with its meaning and definition explained. It is not explained in this article because it's out of its scope. The formula is correct. With proper image rendering you should see something like this. --Tomaxer (talk) 00:40, 9 June 2009 (UTC)


 * It wasn't apparent that the surrounding parentheses had any specific meaning, when they are used just adjacently to indicate grouping. Essentially my annoyance is this: most of the articles on mathematical concepts on wikipedia, even ones on familiar and simple topics like triangular numbers, are often expressed in relatively impenetrable math jargon.  Wikipedia is an encyclopedia for everyone, not just mathematicians.  The articles on various taxonomical groups for example are readable by anyone with a casual acquaintance with biology, or even just animals around them, while the mathematics articles are not.  Consider the article on the binomial coefficient which never answers things how it behaves to a person other than a mathemtician.  How about a pratical example or a brief discussion in what the machinery encapsulates.  In this article, not a single paragraph manages to stay away from spikey jargon.  The first two sentences make an attempt and then quickly descend into using terminology of nth, but even this is half-hearted, since triangular numbers are not dots.  The only part of this page that is readily accessible is the image, which is a shame.  It is a simple topic. JoshuaRodman (talk) 06:04, 10 June 2009 (UTC)

Formula wrong?
Shouldn't:

$$ T_n= 1+2+3+ \dotsb +(n-1)+n = \frac{n(n+1)}{2} = \frac{n^2+n}{2} = {n+1 \choose 2}

$$

actually be:

$$ T_n= 1+2+3+ \dotsb +(n-1)+n = \frac{n(n+1)}{2} = \frac{n^2+n}{2} = n{n+1 \choose 2}

$$

At the top of the article? DMcMPO11AAUK/Talk/Contribs 21:32, 13 October 2009 (UTC)
 * No. See binomial coefficient. The thing at the end with one number over another is not a fraction. As the sentence immediately after the formula in the article explains. —David Eppstein (talk) 21:37, 13 October 2009 (UTC)

Handshake Problem
The 'handshake problem' is not solved by this formula. It would be, if everyone was to shake their own hand. (MKC)

The correct handshake formula is similar see below:



H_n = \frac{n(n-2)}{2} = \frac{n^2-n}{2} $$

— Preceding unsigned comment added by Myesac (talk • contribs) 14:57, 10 January 2008 (UTC)


 * I have removed the statement --Luca Antonelli (talk) 20:25, 14 February 2008 (UTC)


 * I shall reintroduce the comment on "handshake problem". It was correct. Above post and formula is wrong. Number of handshakes with $$n$$ persons is $$T_{n-1}$$. Number of handshakes with $$n+1$$ persons is $$T_n$$. /84.238.113.244 (talk) 22:06, 16 March 2008 (UTC)


 * $$H_n = \frac{n(n-1)}{2} = \frac{n^2-n}{2}$$ is correct for n people shaking hands. Explanation: each person must shake hands with the other n-1 people, resulting in n(n-1).  The factor of two results because of the commutivity of a handshake (e.g. Bob shakes Tom's hand so Tom doesn't have to shake Bob's hand).--Loodog (talk) 02:50, 2 July 2008 (UTC)


 * The link provided next to the sequence of triangular numbers shows a sequence that starts at 0, whereas the sequence on Wikipedia is shown starting at 1. I'm sure that $$T_1 = 1$$ but it seems odd that it starts at zero in the link. The handshake problem would start at zero because one person can't shake hands with themselves. I'm probably just confused, man. Measurements (talk) 21:34, 16 March 2011 (UTC)

Triangular Numbers in Nature
In the Janet Left Step Periodic Table (see second illustration at the Wiki page for Alternative Periodic Tables) periods end with the alkaline earths (helium being similarly s2 electronic configuration here). It had already been determined that every other alkaline earth atomic number (4,20,56,120) was identical to every other tetrahedral number from the Pascal Triangle. The triangular numbers also figure in the periodic relation. Counting leftwards, in the Janet table, from the alkaline earths (including 0 for no movement) always lands you on positions within periods where the quantum number ml=0. —Preceding unsigned comment added by 71.127.246.177 (talk) 05:20, 23 March 2011 (UTC)

A Vote for Zero (who is my hero)
Some authors consider 0 to be a triangular number. Specifically, Sloan's On-Line Encyclopedia of Integer Sequences lists 0 as the first triangular number ( http://oeis.org/A000217 ), and Sloan's is the most common authoritative source for this kind of data. Additionally, the Wikipedia pages for "square triangular number" and "squared triangular number" both indicate that 0 is a triangular number, so there is currently inconsistency in the pages here.

I'm not here to argue for or against the inclusion of 0 (though I admit it seems intuitive to me that it would be as much a triangular number as it is a square number), but if this is not the standard definition we list, it should at least be mentioned that people can and do include 0 in the definition.

There are benefits and detriments to making 0 part of the standard definition. Benefits include consistency with Sloan's resource and definitions, as well as the increased generalizability of triangular numbers to useful situations where 0 is an important element: if 0 is included, then every complete graph (including the "trivial graph" with one vertex and zero edges) contains a number of edges corresponding to the n-th (from zeroth) triangular number. The detriments are mainly the need to reformulate some of the trivia as listed in the "other properties" section: specifically, the "digital root trick", the "ax+b trick", and the "sum of reciprocals" will each need to exclude 0. (I don't think the digital root or ax+b are particularly interesting or mathematically useful, but the reciprocal one is.)

On the other hand, you can extend a few of those results to include 0 now (so every Natural, including 0, is the sum of exactly three triangular numbers, rather than positive integers only are the sum of at most three triangular numbers).

Anyway, the fact that some definitions include 0 should probably be added. (And yes, I am a set theory / CS geek. Very sorry.) --24.26.130.82 (talk) 22:03, 14 June 2011 (UTC)

New proof for sum of two consecutive triangular numbers a perfect square
I have found a new proof for sum of two consecutive triangular numbers resulting into a perfect square : A triangular no.: n(n+1) /2 = n^2+n /2 The next triangular no.: (n+1)^2+n+1 /2 So, sum of two consecutive triangular no.: n^2+n+(n+1)^2+n+1 /2 = n^2+n+n^2+1+n+1/2 = 2(n^2)+2n+2 /2 = n^2+n+1 Since (x+1)(x+1) = (x+1)^2 = x^2+2x+1

Therefore, sum of a triangular no, n, and the next triangular no., results into the square of (n+1). WORTH ADDING, ISN'T IT? 117.226.211.98 (talk) 16:39, 1 January 2013 (UTC)


 * This isn't really any different from the proof in the article, except you've considered the nth and (n+1)th triangular numbers instead of the (n-1)th and nth. Hut 8.5 16:52, 1 January 2013 (UTC)

Triangle triangle numbers?
How much reasearch has be done around the function n=T(1)+T(2)+T(3)+T(4)+T(5)+T(6)...n, the sum of thefirst n triangler numbers? What is the formula for this, and what is the sum of it's recopicals? Robo37 (talk) 14:25, 10 August 2011 (UTC)
 * The sums of the triangular numbers are the tetrahedral numbers. Hut 8.5 14:34, 10 August 2011 (UTC)
 * Oh yeah, how stupid of me. Thanks anyway. Robo37 (talk) 10:40, 11 August 2011 (UTC)
 * Pascal's triangle can be helpful if you want to find n=T(1)+T(2)... and  n=Tetra(1)+Tetra(2).... and so on. 117.226.228.95 (talk) 16:29, 2 January 2013 (UTC)

0 triangular number?
Empty sum says that 0 is a triangular number, well, is it? Moberg (talk) 00:10, 30 November 2011 (UTC)
 * Yes. Anders Kaseorg (talk) 23:06, 3 August 2012 (UTC)
 * Is 0 the 0th triangular number or the 1st triangular number? 117.226.228.95 (talk) 16:30, 2 January 2013 (UTC)
 * It's the 0th. Plug n=0 into the formula given in the article and you get 0. Hut 8.5 19:33, 2 January 2013 (UTC)

An elegant property#2 and #3
If n is a triangular no [I will refer to them as tri no. from now], then 25n + 3 is also a tri no! Lets see how:

25((n^2+n)/2) + 3 = 25n^2 + 25n + 6 /2= (25n^2 + 20n +4) + (5n + 2) /2. Now: (25n^2 + 20n +4) + (5n + 2) /2 or, (25n^2 + 10n + 10n +4) + (5n + 2) /2 or, ((5n(5n + 2) + 2(5n + 2)) + (5n + 2)/2 or, (5n + 2)(5n + 2) + (5n + 2)/2 or, (5n + 2)^2 + (5n + 2)/2 Wow! We found out that every nth tri no. when multiplied by 25 and added 3, it produces the (5n + 1)th tri no! For eg: 1st tri no: 1*25+3 = 28 = 7th tri no!

If n is a triangular no, then 8n + 1 is a square no! Lets see how:

8((n^2+n)/2) + 1 = 8n^2 + 8n + 2 /2= 4n^2 + 4n + 1. Now: 4n^2 + 4n + 1 or, 4n^2 + 2n + 2n + 1 or, 2n(2n+1) + 1(2n+1) or, (2n+1)^2 Wow! We found out that every nth tri no. when multiplied by 8 and added 1, it produces the (2n + 1)th square no! For eg: 1st tri no: 1*8+1 = 9 = 3rd(1*2+1) square no!  Note: Since, (2n+1)^2 = 4(n^2 + n) + 1, the square produced would always a odd, as would be its square root. 117.226.159.194 (talk) 11:59, 15 January 2013 (UTC)

An elegant property#4
Suppose n is a triangular no [I will refer to them as tri no. from now]. The sum of squares of two tri nos is a tri no! Lets see how:

If n is a tri no. then Trin-1 + Trin: ((n-1)2 + (n-1) /2)2 = ((n2 - 2n + 1) + (n - 1)/2)2 = n2 - n)2 /22 = n4 + n2 -2n3 /4 Much the same for: n2 + n /2 results to n4 + n2 + 2n3 /4 (n4 + n2 -2n3 /4) + (n4 + n2 +2n3 /4) = 2n4 + 2n2 /4 = n4 + n2 /2 = (n2)2 + n2 /2

Wow! We found out that every (n-1)th tri no. and (n)th tri no are added, it produces the (n2)th tri no! For eg: 1st tri no + 2nd tri no.: 12 + 32 = 10 that is 4 [22]th tri no! 117.226.159.194 (talk) 13:31, 15 January 2013 (UTC)

An elegant property
Not sure if this is worth putting in, but a nice fact. Form a sequence of numbers, starting from one, then repeatedly multiply by nine and add one to get the next term. All the numbers in this sequence are triangular. —Preceding unsigned comment added by 82.6.96.22 (talk) 18:00, 3 May 2011 (UTC)


 * I found a proof! The result is always 3n+1 th triangular number! 115.250.59.221 (talk) 11:50, 5 January 2013 (UTC)


 * Might be worth mentioning in the article, but only if we can find a reliable source for it. —David Eppstein (talk) 17:04, 5 January 2013 (UTC)


 * Did not have time to publish it then, but now here it goes: [looks my IP has changed]
 * Basic formula = (n^2 + n)/ 2
 * Now what we want to say is that: 9(n^2 + n/ 2) + 1 =  9n^2 + 9n + 2/ 2 is also in the form of n^2 + n/ 2. Lets see:
 * 9n^2 + 9n + 2/ 2 = 9n^2 + 6n + 1 + 3n + 1/ 2 = (9n^2 + 6n + 1) + (3n + 1)/ 2 = (3n +1)^2 + (3n + 1)/ 2
 * Wow! if we plug in 3n + 1 in the formula we get the result! For example 1st triangular no. = 1 therefore (3*1)+1 = 4th triangular no. must be (9*1)+1 = 10, which it is! 117.227.200.134 (talk) 17:45, 13 January 2013 (UTC)


 * Fine, but we need a reliable source before we can add it to the article; otherwise, it is original research. —David Eppstein (talk) 18:40, 13 January 2013 (UTC)


 * What the hell? We need to add informative content and blah blah, for which it is qualified. Secondly, we should add information which is provably true or has some sources to rely upon, which it also qualifies. Why do we need to hesitate to add it, when it comfortably with the basic rules. And still if you are too strict for rules, there exists a godfather rule called, WP:Ignore all rules. 117.226.193.247 (talk) 14:02, 14 January 2013 (UTC)


 * It seems likely to me that in this case the sources really do exist. So you just need to find them. —David Eppstein (talk) 15:51, 14 January 2013 (UTC)


 * Found them:  ! The rest lies with you as I don't know LATEX, hope you do it, I am going to bed. 117.226.193.247 (talk) 16:35, 14 January 2013 (UTC)


 * Flipping through the feedback, I really consider one reader's demand of the first 100th tri. no. in the article, much like it is in prime number and other such articles. [And I think the article needs expansion with more helpful illustrations.] What do you think? 117.226.193.247 (talk) 16:39, 14 January 2013 (UTC)


 * These numbers are the repunits in base 9; see, which notes that they are all triangular numbers. —David Eppstein (talk) 16:36, 15 January 2013 (UTC)

An elegant property #5
n is equal to $$\frac{\sqrt{8x+1}-1}{2}$$, such that $T_{n} = x$! Let's see how:

First, the basic formula:
 * n(n+1) / 2 = (n2 + n) / 2 = x
 * = 8x = 4(n2+n) = 4n2+4n [Multiplying by 8 on both sides]
 * = 8x + 1 = 4n2 + 4n + 1 [Adding 1 to both sides]
 * = 8x + 1 = 4n2 + 2n + 2n + 1
 * = 8x + 1 = 2n(2n + 1) + 1(2n + 1)
 * = 8x + 1 = (2n + 1)(2n + 1) = (2n + 1)2
 * = $$\sqrt{8x+1} = 2n + 1$$ [Square root of both sides]
 * = $$\sqrt{8x+1} - 1= 2n$$ [Subtracting 1 to both sides]
 * = $$\frac{\sqrt{8x+1}-1}{2}= n$$ [Dividing 2 on both sides]

Bingo! Let try it with an example, where x = 10:
 * $$n = \frac{\sqrt{8x+1}-1}{2} = \frac{\sqrt{80+1}-1}{2} = \frac{9-1}{2} = 4$$

Yoho! The 4th tri. no. is 10, as we found out! 117.227.149.197 (talk) 14:06, 28 January 2013 (UTC)


 * Yes, this is just what you get if you apply the quadratic formula to the formula for triangular numbers. Hut 8.5 18:37, 28 January 2013 (UTC)

Alternate notation
Does anyone else support the inclusion of other, perhaps less common, forms of notation? I was able to dig up discussion here as well as some additional support for the n? notation here. 67.171.222.203 (talk) 22:53, 15 April 2014 (UTC)

Overcomplicated Result notation
Please, could you edit the result formula, removing the extremely unpopular "binomial coefficient" by the more popular n * (n+1) / 2 "as the Last Result" ? if you want, you can add the reference to the binomial coefficient in a side note, because when the vast majority of the people read "open parenthesis" (n+1) / 2 "close parenthesis", they will just discard the parenthesis because it means nothing when used as the first and last characters in algebraic notation (which will be the CONTEXT employed to read this result, since up to that point, just simple algebra was used. Please, take this info into account (if the objective is give people answers instead of letting them confused, trying to find why the "n" was dropped in the final result, it is just "(n+1)/2" ? -- Please.

Popular Culture
Personally, I wouldn't consider the bible being "pop culture" — Preceding unsigned comment added by 211.29.185.4 (talk) 06:08, 15 June 2007 (UTC)


 * Yeah, it’s religious, and been around for hundreds of years. So saying that it’s popular culture doesn’t make sense. — Preceding unsigned comment added by 2601:140:8a00:2059:fde6:9b72:9a53:1d4 (talk) 13:01, 1 March 2019 (UTC)
 * Not personally, but as a matter of fact, the Bible is the most published book in history. Also, using logic, a book that is the "most published book" is automatically considered "popular culture" (unless one wants to do "freestyle Logic"). Drout 0 (talk) 19:57, 6 August 2022 (UTC)

Exponential functions of Triangular numbers
These formulas apply to any x > 0

T(x2) = T(x-1)2 + T(x)2 T(1) = 02 + 12 = 1 T(4) = 12 + 32 = 10 T(9) = 32 + 62 = 45 PROOF T(x2) = [x2 * (x2 + 1)] / 2 = [x4 + x2] / 2 T(x-1)^2 = [x4 - 2x3 + x2] / 4 T(x)^2 = [x4 + 2x3 + x2] / 4 T(x)2 + T(x-1)2 = [2x4 + 2x2] / 4 = [x4 + x2] / 2 edit: Whoops, just noticed that this one is redundant. My bad. Next, any T(x)2 is equal to the sum of cubes from 1 to x T(1)2 = 13 = 1 (12) T(2)2 = 13 + 23 = 9 (32) T(3)2 = 13 + 23 + 33 = 1 + 8 + 27 = 36 (62) 1 + 8 + 27 + 64 = 100 (102) I have not found a formula to express this (as is expected with iterative functions) but it can be verified in about 20 lines of java code. Finally, 6*T(x) + 1 = (x+1)3 - x3 23 - 13 = 8 - 1 = 7 = 6 * T(1) + 1 33 - 23 = 27 - 8 = 19 = 6 * T(2) + 1 PROOF (x+1)3 = x3 + 3x2 + 3x + 1 -x3 = 3x2 + 3x + 1 6 * T(x) + 1 = 6 * (x2 + x) / 2 + 1 = 3x2 + 3x + 1 Hopefully these can be put to use. They seem to be similar in structure to existing algebraic formulas, but a bit more straightforward. Jozbornn (talk) 06:50, 2 March 2015 (UTC)
 * The sum of cubes fact that you state is well known — see squared triangular number. But if you want any of this to be incorporated into the article (and why else would you be posting it here?) you'll need to find reliable sources that already state these facts — otherwise it would be original research. —David Eppstein (talk) 07:24, 2 March 2015 (UTC)

Assessment comment
Substituted at 02:39, 5 May 2016 (UTC)

Easy formula
Why isn't there a easy formula? Like f(x)=(x/2)(x+1) works very well. 178.121.66.129 (talk) 17:39, 11 March 2019 (UTC)
 * That's the second-to-last of the terms in the big formula at the start of the "Formula" section. It's also repeated a couple more times in the same section. —David Eppstein (talk) 17:56, 11 March 2019 (UTC)
 * It is a little confusing, because it looks like just one big equation, rather than three different ways to write the same thing. Sometimes we put them all on separate lines, but perhaps its just done this way to save space since two of the three identities are just one term. — Soap — 03:45, 7 April 2020 (UTC)

Easier proof
Is there a reason why there isn't an easier proof on the page? I mean, it can be solved by anyone who saw $$\sum_{x=1}^n x = n(n+1)/2$$ in school; by induction.

Obviously 1³ = 1². if applies for $$n$$, then it applies for $$(n+1)$$ as well; so


 * $$( \sum_{x=1}^{n+1} x )^2 = \sum_{x=1}^{n+1} x^3 $$

seperate $$n+1$$


 * $$( ( \sum_{x=1}^n x ) + (n+1))^2 = ( \sum_{x=1}^n x^3 ) + (n+1)^3$$

considering we assume it applies for $$n$$, we can substitute $$\sum_{x=1}^n x^3 = ( \sum_{x=1}^n x )^2$$


 * $$( ( \sum_{x=1}^n x ) + (n+1))^2 = ( \sum_{x=1}^n x )^2 + (n+1)^3$$

solving the quadratic equation


 * $$( \sum_{x=1}^n x )^2 + 2(\sum_{x=1}^n x)(n+1) + (n+1)^2 = (\sum_{x=1}^n x)^2 + (n+1)^3$$

remove $$(\sum_{x=1}^n x)^2$$ from both sides


 * $$2(\sum_{x=1}^n x)(n+1) + (n+1)^2 = (n+1)^3$$

considering $$(\sum_{x=1}^n x) = n(n+1)/2$$


 * $$2n(n+1)(n+1)/2 + (n+1)^2 = (n+1)^3$$

as $$2*a*b*b/2 = a*b^2$$


 * $$n(n+1)^2 + (n+1)^2 = (n+1)^3$$

next, we single out $$(n+1)^2$$ as $$a*b^2 + b^2 = b^2 (a+1) $$


 * $$(n+1)^2(n+1) = (n+1)^3$$

which proves the theorem

Qube0 (talk) 17:18, 5 January 2020 (UTC)


 * Easier than what? Other than a graphical illustration, there's no proof in this article at all, which is perfectly reasonable.  Inductive proofs of identities like this aren't generally very enlightening and just tend to clutter up the article.  On a side note, your proof is full of errors – most seriously, starting by writing what you're trying to ultimately prove, rather than winding up with that as the final result. –Deacon Vorbis (carbon &bull; videos) 17:44, 5 January 2020 (UTC)


 * I did find the existing proof by induction here rather hard to follow, so I tried to make it easier to understand, based on a short proof in Spivak's lovely Calculus. He presents his proof as the first "real" proof by induction in the book, and in the context of introducing the reader to induction in general, so I think this particular article is actually a great place to give a proof by induction. I wrote it out in an especially explicit style, such that someone who's never used induction could hopefully follow it, since I do think it makes a great introduction to induction as a proof strategy. Mesocarp (talk) 00:53, 27 October 2022 (UTC)

non-integer triangular numbers
Since the $$ \frac{n(n+1)}{2}$$ formula can be evaluated for all real numbers, both positive and negative, integer or non-integer, is there any special term for the numbers that lie along the resulting graph? x^2 evaluates to the same value regardless of sign, so the resulting curve is symmetrical across the axis of x=-1/2. But Im more interested in the fractional terms and if there is anything special about them. e.g. the πth "triangular number" is  approximately 6.50559852734. If there's anything to this, it certainly would be worth mentioning, but I couldnt find anything about "generalized triangular numbers" anywhere. Thank you, — Soap — 03:42, 7 April 2020 (UTC)

Proposed merge from Termial
Termials and triangular numbers are the same thing. Since Wikipedia articles are about concepts not names (WP:NOTDICT), they should have together only one article. Therefore, I am proposing a merge. I think that termial is also a rather obscure term for what is much more widely known as a triangular number, so the merge should go from termial to triangular number. Any other opinions? —David Eppstein (talk) 19:26, 20 August 2020 (UTC)
 * Support I wouldn't say they are the same thing. One is a sequence while the other a series, and termial can apparently apply to non-integers. However, they are certainly intertwined, and Termial might be better as a section of the Triangular Number article. The term "termial" doesn't seem to appear in much mathematical literature though, so I question its notability in the first place. Kstern (talk) 20:54, 3 September 2020 (UTC)
 * Support They're the same thing! If someone wants to know more about termials, and they read the article about triangular numbers, that will serve their purpose just fine. — Preceding unsigned comment added by 73.241.189.0 (talk) 17:00, 24 September 2020 (UTC)


 * Support it doesn't make much difference whether the concept is presented as a sequence or a series, the two are clearly functionally equivalent and this article does already note that triangular numbers can be represented as a series. Triangular number is definitely the primary usage and I suggest we just note somewhere that they are sometimes known as termials (The Art of Computer Programming is very influential). Termial doesn't actually specify how it can be calculated for non-integers.  Hut 8.5  19:11, 24 September 2020 (UTC)
 * This MUST be merged with triangular numbers. It has no independent conceptual difference, and much less content, and it's only used by Knuth, while triangular number has been used since the pythagoreans at least
 * Termial means exactly the same as triangular number. The idea is to simplify knowledge, not to create for each synonim a different new article. And this is obviously a term only used by no one.
 * Santropedro (talk) 03:13, 4 January 2021 (UTC)

Seeing support and no opposition, I have gone ahead and done the merge. —David Eppstein (talk) 05:08, 4 January 2021 (UTC)

Rotations in n-dimensional geometry
The amount of rotations in any n-dimensional geometry is the triangular number of n-1. Unfortunately, I haven't found anything on this on the internet so it would count as OR. Is there any way to get around this?

You can play around with it yourself if you want: https://www.mathgoodies.com/calculators/triangular-numbers

75.129.231.4 (talk) 23:19, 28 October 2020 (UTC)


 * What exactly do you mean by "amount of rotations?" Do you mean possible axes of rotation?  In three dimensions an infinite number of rotational axes exist.—Anita5192 (talk) 00:00, 29 October 2020 (UTC)

Edward Waring
While looking at Edward Waring's Meditationes Algebraicae (3rd ed, 1782) for another purpose, I noticed that he states (in Latin, of course), without proof, that every whole number is the sum of 1, 2 or 3 triangular numbers. (This doesn't exclude the possibility that it is also the sum of 4, 5, 6... etc such numbers). (op. cit. p 349 prop. 3.) Waring goes on to state the corresponding propositions for pentagonal and hexagonal numbers (prop. 4) and squares (prop. 5). (This is on the same page as the proposition often known as Waring's Theorem.) I was wondering if this was already known before Waring. Looking at the article here, I see that it was later rediscovered by Gauss, who was excited enough to call it his Eureka theorem. However, I see that it is a special case of a conjecture of Fermat, known as his Polygonal Number Theorem, but characteristically stated without proof. Quite possibly Waring found it in Fermat. As a conjecture, without proof, it is therefore certainly earlier than Gauss, though Gauss may have been the first to prove it for the triangular case.2A00:23C8:7907:4B01:6415:B77:5411:4DE (talk) 14:06, 17 June 2022 (UTC)