# Talk:Arithmetic progression

WikiProject Mathematics (Rated Start-class, Mid-importance)
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.
Mathematics rating:
 Start Class
 Mid Importance
Field: Number theory
One of the 500 most frequently viewed mathematics articles.
WikiProject India (Rated Start-class, Low-importance)
Start  This article has been rated as Start-Class on the project's quality scale.
Low  This article has been rated as Low-importance on the project's importance scale.

## Merging with arithmetic series

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)
Fredrik, thanks! Oleg Alexandrov 19:35, 20 August 2005 (UTC)
No problem :) - Fredrik | talk 19:38, 20 August 2005 (UTC)

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 (talkcontribs)

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)

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)

## Product

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)

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

## sum of sine and cosine in arithmetic progression

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

### Sum of Sines

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

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

### Sum of Cosines

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)

## History

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