Talk:Arithmetic progression

This  level-5 vital article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

Mathematics Mid‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Mid This article has been rated as Mid-priority on the project's priority scale. India Low‑importance This article is within the scope of WikiProject India, which aims to improve Wikipedia's coverage of India-related topics. If you would like to participate, please visit the project page.IndiaWikipedia:WikiProject IndiaTemplate:WikiProject IndiaIndia Low This article has been rated as Low-importance on the project's importance scale. This article was last assessed in April 2012.

According to Mathworld, which has a link in the article, an arithmetic series is the sum of an arithmetic progression or sequence. Charles Matthews has obscured this distinction by redirecting arithmetic series to arithmetic progression. I'm not sure whether the distinction made by Mathworld is commonly recognised by mathematicians, so I'm not going to revert the change. I'll wait for comments. -- Heron 15:12, 7 Mar 2004 (UTC)

The redirect only implies that information on arithmetic series is contained in the arithmetic progression article, not that the two terms are synonymous. The article is quite clear on the latter point (and mathworld is right, of course). -- Arvindn 15:39, 7 Mar 2004 (UTC)

Yes, the article has both definitions; I don't see anything obscure about it.

Charles Matthews 15:59, 7 Mar 2004 (UTC)

How about merging series&progression articles as it's done with the geometric series&progression?

I strongly think arithmetic series should be merged back into this article. Fredrik | talk 14:20, 19 August 2005 (UTC)Reply

Fredrik, thanks! Oleg Alexandrov 19:35, 20 August 2005 (UTC)Reply

No problem :) - Fredrik | talk 19:38, 20 August 2005 (UTC)Reply

Mathworld includes several Egyptian math entries. A small number are my own. My view is Mathworld editors stress modern number theory conversions of rational numbers to non-concise unit fraction series, often to awkward versions of the greedy algorithm, thereby being of little value to new readers wanting to know how to read the historical Egyptian fraction rational numbers, and associated formulas. —Preceding unsigned comment added by Milogardner (talk • contribs)

Does this have anything to do with the contents of our article on arithmetic progressions? If not, it shouldn't be here. —David Eppstein (talk) 19:02, 16 September 2010 (UTC)Reply

Of course it does. The first known arithmetic progression in the Western Tradition was written in the Kahun Paprus around 1900 BCE and again in the Rhind Mathematical Papyrus in problems 40 and 64. The formulas that found the largest and smallest terms in two different arithmetic progressions were algebraically related, and not algorithmic. The two formulas looked very much like Gauss' childhood story of summing seccessive additions of 1 to 100 by finding 50 pairs or 101, obtaining 5050. Egyptians did much better. Milogardner (talk) 19:26, 16 September 2010 (UTC)Reply

I toyed with the idea of taking the product of an arithmetic progression, and came up with the following expression (initial term a, common distance s, and n terms):

${\displaystyle s^{n}\times {\frac {\Gamma \left(a/s+n\right)}{\Gamma \left(a/s\right)}}}$

Anyone seen this before, and is it useful? - Fredrik | talk 16:38, 19 August 2005 (UTC)Reply

Interesting. This seems to be generalizing the formula 1·2··· n=Γ(n+1). I hever saw it before. I quite don't know what one would use it for though.

This article is rather stubby, any additions to it, like your formula, would be welcome.

By the way, what do you think of merging this article with Arithmetic series? Both are stubby and don't talk about much different things? Oleg Alexandrov 19:09, 19 August 2005 (UTC)Reply

Yeah, it's derived from that formula.

It could be useful in numeric computation, to obtain the product (or its logarithm) of an immensely long progression in O(1) time, though I'm not sure in what kind of context you'd need to do that.

There is also an obvious problem, that it is invalid when a/s is a negative integer (though for computations that could be handled easily as a special case).

As stated a couple of paragraphs up, yes, I think merging would be a good idea. Fredrik | talk 19:35, 19 August 2005 (UTC)Reply

So we arrived independently to the same conclusion. I will merge the articles soon if I don't forget. If you get to it before me, that will be fine too. Oleg Alexandrov 03:53, 20 August 2005 (UTC)Reply

Actually, seems like I'd just rediscovered the Pochhammer symbol, heh. Fredrik | talk 13:25, 23 October 2005 (UTC)Reply

I think this would be useful to add under 1 Sum (arithmetic series)

The arguments of a sum of sines can be in arithmetic progression, as follows

${\displaystyle S=\sin \varphi +\sin {(\varphi +\alpha )}+\cdots +\sin {(\varphi +n\alpha )}}$.

It has also, like a normal arithmetic sequence, a concise formula, written as

${\displaystyle S={\frac {\sin {({\frac {(n+1)\alpha }{2}})}\cdot \sin {(\varphi +{\frac {n\alpha }{2}})}}{\sin {\frac {\alpha }{2}}}}}$.

Analogous to the sum of sines with their arguments in arithmetic sequence, there is also one with cosines:

${\displaystyle S=\cos \varphi +\cos {(\varphi +\alpha )}+\cdots +\cos {(\varphi +n\alpha )}}$.

There is also, the general expression, which is somewhat similar to the one of the sines:

${\displaystyle S={\frac {\sin {({\frac {(n+1)\alpha }{2}})}\cdot \sin {(\varphi +{\frac {n\alpha }{2}})}}{\sin {\frac {\alpha }{2}}}}}$

${\displaystyle S={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \cos {(\varphi +{\frac {n\alpha }{2}})}}{\sin {\frac {\alpha }{2}}}}}$.

In response to the unsigned material above: I think this material represents a "trigonometric progression" and not an "arithmetic progression". Maybe there is someplace it can be placed in the trigonometry articles. Thelema418 (talk) 05:06, 23 August 2012 (UTC)Reply

For history of AP in old Indian texts, see this research paper.--Nizil (talk) 20:10, 1 January 2017 (UTC)Reply

If the article is going to cover history at all, it should also mention Archimedes, who proved an equivalent formula for the sum of an arithmetic progression as a part of his more difficult proof of the sum of the squares of the terms of the progression, in his treatise 'On Conoids and Spheroids' (around 300 BC). This is mentioned in T. L. Heath's edition. I would guess that the formula was also known in some form to the Arab mathematicians, but I have not researched the history of the subject in depth.109.150.6.229 (talk) 19:52, 19 March 2019 (UTC)Reply

There needs to be a separate article named common difference of an arithmetic progression. Huzaifa abedeen (talk) 05:14, 1 October 2020 (UTC)Huzaifa abedeenReply

I've trimmed a moderate amount from this article, culling stuff like bullet points that restate the obvious and look like they were copied out of a study guide for a junior-high quiz, and a "section" devoted to two lines of Python. Wikipedia is not a textbook, a cheat sheet for elementary formulae, or a Programming 101 manual. In addition, claims like Arithmetic Progression was invented by Johann Carl Friedrich Gauss. are blatantly unhistorical and even make the article self-contradictory. Pop-math wiki websites and random study guides are not sources we should depend upon, particularly when standard texts and peer-reviewed papers on the history of mathematics are plentiful. XOR'easter (talk) 20:03, 29 October 2020 (UTC)Reply

Please consider adding common difference ${\displaystyle d}$, general form of an AP ${\displaystyle a,a+d,a+2d,a+3d,...}$, general term of an AP ${\displaystyle a_{n}}$, middle term of an AP, and Arithmetic mean. Huzaifa abedeen (talk) 05:42, 31 October 2020 (UTC)Reply

Why is there not one bit of content discussing the notion of an infinite version of the arithmetic series, if only to note that nontrivial ones are divergent? There needs to be a mention of this so that there isn't a complete void in information regarding this topic, along with talking about various summation methods to give defined values to normally divergent series.

No really, why is the word "infinite" completely absent when talking about arithmetic series? Even after the link infinite arithmetic series (in the series and sequence template box, in the divergent series section together with a bunch of examples of divergent series with their summation method results) became nothing but a redirect to this article? 2600:1012:A021:7A65:AD81:13C9:15D2:E5DF (talk) 02:52, 17 May 2025 (UTC)Reply

To further clarify, the current description of "arithmetic series" only talks about the finite version, as if the term is defined as a finite arithmetic sum (contrary to the usual meaning of a series, where the default is to refer to the infinite version). This is inaccurate; an infinite version of an arithmetic series is mathematically valid. Geometric series article talks about both the infinite series, the primary version, as well as the finite geometric series.

So why not have that for arithmetic? Most are divergent series, but divergence doesn't stop making them valid series. And it is precisely this that is a good starting point for talking about series such as 1 + 2 + 3 + 4 + ... and their summation methods. 2600:1012:A021:7A65:AD81:13C9:15D2:E5DF (talk) 14:54, 17 May 2025 (UTC)Reply

The article could also use a sigma notation expression for the finite and infinite series. 2600:1012:A024:A5C0:5540:B7D8:4D6A:270A (talk) 06:07, 19 May 2025 (UTC)Reply

Anyone else watching the edits here and to the article should be aware that discussion has mostly been taking place at Talk:1_+_2_+_3_+_4_+_⋯#Why is the word "arithmetic" nowhere mentioned here?. --JBL (talk) 20:19, 19 May 2025 (UTC)Reply