# Talk:Geometric progression

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## A question

Can common ratio's be imaginary? Maybe we should include that since the article specifically says they can be negative. —Preceding unsigned comment added by 72.229.188.248 (talk) 02:34, 28 January 2011 (UTC)

## Paragraph problem

I don't really agree with the paragaph:

One cannot see why the proportion called arithmetical is any more arithmetical than that which is called geometrical, nor why the latter is more geometrical than the former. On the contrary, the primitive idea of geometrical proportion is based on arithmetic, for the notion of ratios springs essentially from the consideration of numbers

That seems a rationalisation based on number, but it is not certain that mathematics started as number.

It is "not certain" that it did? Indeed, isn't it certain that it did not? The notion of real number grew out of geometry!

For example, take a square, double the length of its sides, double again, and again. Clearly the side lengths are in geometric progression; so too are the areas. Take a different square, add 2 to the length of the sides (2 what?), add 2 again, and again. This time the side lengths are in arithmetic progression, but the areas are not. It seems natural to call the former geometric, leaving arithmetic to the latter. In the medieval quadrivium, arithmetic was pure number, geometry was number in space, music number in time, and astronomy number in space and time; but I doubt that was the order in pre-history.--Henrygb 13:13, 21 Mar 2004 (UTC)

Both Geometric sequence and Geometric series deal with the other, so i"m putting them together under Geometric progression and redirecting.

## progression, series or sequence?

Why isn't this article called "geometric sequence" but "geometric progression"? I don't find it clear what is meant with progression. --Abdull 13:43, 19 Jun 2005 (UTC)

a sequence means that the terms are just followed on after the other. a series means that every term is summed to the next. a prograssion just means that the numbers progress, keep going, withut specifyin if the terms are added or not
Cako 20:49, 2 November 2006 (UTC)goldencako

Sequences are multiples or powers of a quantity (eg 'r'), shown as a list of terms separated by COMMAS (a,ar,ar2, etc in example). Series show the SUM of those terms (a + ar + ar2 + ...). 'r' or 'x' is fixed for the calculation of a single series, but varies in practice as the top right hand diagram illustrates, so calling it a fixed number is a bit misleading. When I did Maths 2T as part of my Chemical Engineering degree one of the recommended texts was a little book called 'Sequences and Series' (part of the RKP series of maths texts). It all seemed rather dry at the time. Had it been introduced by way an example such as Arrhenius' greenhouse law (see http://en.wikipedia.org/wiki/Svante_Arrhenius#Greenhouse_effect , and upcoming BBC series) it would have been much more intriguing. Geometric series represent exponential growth or decay, as represented by population dynamics (Malthus) or climate change (Arrhenius). I think that the geometric series explanation should be retained to illustrate the relationship between sequences and series, and the difference between arithmetic (LINEAR) progressions/series and geometric (EXPONENTIAL, ie POWER) progressions. Merrison. Merrison (talk) 18:15, 23 November 2015 (UTC)

The example for return on capital interest is not correct and misleading. Interest rates are calculated using compound interest rates. If one could get $16000 out of$2000 in 6 years, that would be quite awesome. I am not familiar with wikipedia to fix this example, thanks for taking note of it. --Vastinnocentaims 15:20, 16 August 2005 (UTC)

## Mistake in ratio?

I think the exponent in the ratio (second image under Formulae) should be 1 / (n - 1) instead of just n - 1. Anyone can verify this? Aggelos Orfanakos 00:09, August 30, 2005 (UTC)

## missing formula

it appears that the current version of the page (on my display)is missing the first formula, namely the mathethematical definition of the sequence itself, before going into the definitions of the scale factor and common ratio

It's been a few years since I studied this subject, so I came on here to help myself solve a little problem I was having, and found that this article was of little help. Can someone please clarify the geometric series part of the article? What is X? I can tell ${\displaystyle X^{n}}$ is the Nth term of the geometric sequence, but what is X? Maybe put an example in?--149.135.21.11 14:33, 13 March 2006 (UTC)

## Notation

${\displaystyle a_{n}=a\,r^{n-1}\quad {\mbox{where n is an integer such that }}n\geq 1}$

In the case of the above, would it not be more elegant to say "${\displaystyle n\in \mathbb {N} }$" in place of "${\displaystyle {\mbox{where n is an integer such that }}n\geq 1}$"?

I'd change it myself, but I wouldn't want to mess around with something if there's a specific reason for that form. --MightyPenguin 11:53, 20 March 2006

${\displaystyle \mathbb {N} }$ can contain zero, depending on the circumstances (see natural number). As it's slightly ambiguous, it's probably better to leave it as is. siafu 21:44, 2 May 2006 (UTC)

## Credit for Series Simplification

Who is to be given credit for discovering the geometric series simplification? Gauss? 131.120.10.130 19:34, 19 July 2006 (UTC)

## Product

Can anyone add a subsection of the product of a geometric sequence?

Hey, I was wondering if it would be appropriate to add this to the article: Why do we call it a Geometric Series? If someone knows the answer, please add this here.

## Product

The product of a geometric sequence can be expressed as

${\displaystyle P=\prod _{i=0}^{n-1}ar^{i}}$

However, it can also be expressed using only the first and last termes and the total number of terms.

${\displaystyle P=(a_{1}\cdot a_{2})^{n/2}}$

Which leads to the interesting conclusion, that

${\displaystyle {\sqrt[{n}]{P}}=a_{\frac {(n+1)}{2}}}$.

If so, should i also include some proofs?
Cako 20:48, 2 November 2006 (UTC)goldencako
A danger for the first claim above is when n is odd, in which case the question arises as to what ${\displaystyle x^{1/2}}$ is for negative or complex ${\displaystyle x}$. E.g. consider the case where ${\displaystyle n=1}$ and ${\displaystyle a_{1}=-1}$.
As for the second claim, it is evidently only defined for the case that n is odd. Even then, it doesn't hold if r is allowed to be non-real: suppose that ${\displaystyle n}$ is 3, then multiplying ${\displaystyle a}$ by cis(2π/3) doesn't change P but does change each term ${\displaystyle a_{i}}$ (including ${\displaystyle a_{\frac {(n+1)}{2}}}$). The claim might hold if ${\displaystyle n}$ is odd and ${\displaystyle {\sqrt[{n}]{P}}}$ is read as ‘one of the ${\displaystyle n}$'th roots of P’ instead of the usual ‘the principal ${\displaystyle n}$'th root of P’. (See Nth root.)
More generally: The principles behind [[WP::No original research]] may apply to such derivations: It's good to cite a reliable external source for such derivations, to give more confidence that the corresponding proof has been checked by an expert. (Purely-mathematical deductions are something of a boundary case for the no-original-research principle; e.g. [[WP::Attribution]] says that “straightforward mathematical calculations” don't count as original research.)
Pjrm 13:11, 1 April 2007 (UTC)

## Interesting

I have someone following me around deleting my comments. Anyone want to comment on the following? --JohnLattier 23:06, 3 November 2006 (UTC)

- ${\displaystyle \sum _{k=0}^{\infty }ar^{k}=\lim _{n\to \infty }{\sum _{k=0}^{n}ar^{k}}}$ - - - Remember, the infinite sum only converges towards the limit. The limit represents the number to which the sum is convergent. The equal sign is improper. --JohnLattier 05:52, 3 November 2006 (UTC) - - I suggest: - - ${\displaystyle \sum _{k=0}^{\infty }ar^{k}\approx \lim _{n\to \infty }{\sum _{k=0}^{n}ar^{k}}}$ - --JohnLattier 05:55, 3 November 2006 (UTC)

I believe it's inappropriate to make such a response to a proposed change in Wikipedia without giving at least some evidence for your position.
(In this case, some evidence is that the Infinite series article claims that the ${\displaystyle \sum _{n=0}^{\infty }}$ notation specifically means the limit of the sequence of partial sums. Granted, citing a Wikipedia article is not particularly strong evidence for what should go in another Wikipedia article, but it's considerably better than nothing.)
Even with strong evidence, it is not “end of story”, because opposing evidence can be given. (E.g. in the current case, the dispute is merely about the meaning of a notation, and one notation can sometimes have different meanings. Here, the meaning in the case of divergent series in particular has varied over time, and may well continue to evolve. Maybe the exact proposed change should not be adopted, but maybe some aspect of such an objection could reasonably be incorporated into the relevant article.)
If one disagrees with a proposed change, then one's own conviction does count as evidence, but is to be set against the opposing evidence that the proposer apparently believes that the proposed change is a good one. It suffices to give the evidence one has on hand (“I am confident that the proposed claim is false”, possibly giving some reason for the person to trust your judgement if you think that they too are confident of their position). I understand that one may strongly want to prevent certain changes, but I believe that the response above tends to have the undesirable side effect of dissuading future constructive changes.
Pjrm 14:50, 1 April 2007 (UTC)

## Constant Factor

Why is there a constant factor "a" in all of these formulae? Isn't it more concise to just say ${\displaystyle \sum _{k=0}^{\infty }r^{k}={\frac {1}{1-r}}}$ instead of :${\displaystyle \sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}}}$ ? Dougthebug 23:36, 5 February 2007 (UTC)

Ans: Usually when deriving a formula you use a general form of the equation. You just can't assume things like a = 1. You need to derive equations using generic values. Sachhidh

I absolutely agree that the a is redundant, and should be removed - it adds nothing to the mathematics and anyone who is well-studied enough to be doing finite sums should be able to extract an a from both sides. I feel like it makes the mathematics clumsy and should not be in an encyclopedia. —Preceding unsigned comment added by 165.124.212.240 (talk) 06:10, 30 September 2009 (UTC)

The a is there because the definition of geometric progression does not assume that it starts with 1. It's probably simpler to just keep the a than to keep having to explain why it was taken out.--RDBury (talk) 13:20, 1 October 2009 (UTC)
The article deals with geometric series in general. I cannot see any good reason for giving anything less than the most general formulas. It is true that it is easy to derive the general formulas from more specific ones, but why is that a justification for not giving the formulas in their most general form? JamesBWatson (talk) 18:49, 4 October 2009 (UTC)
In addition to the above math-based reasons to retain the a, here's a encylcopedia-based suggestion: discussion of these series is taught in high school with the constant a (sometimes at least). So retaining the a makes the page more useful to readers with that level of experience. Citations for the assertion about math teaching include the "Kaplan SAT Subject Test Mathematics Level 2" 2010(?) edition. The constant was included in my daughter's high school classes, although that hardly qualifies as a decent citation. DavidHolmes0 (talk) 21:35, 6 September 2010 (UTC)

## Proof that 1 + 2 + 4 + 8 + ... = -1

n = 1 + 2 + 4 + 8 + 16 + ...

2n = 2 + 4 + 8 + 16 + 32 + ...

n = 1 + 2n

0 = 1 + n

n = -1

I talked to a mathematician about this proof and he didn't see any flaw in it. He said that the only thing that it proves is that if the series does converge, that it would converge to -1. An example of where it would converge to -1 would be in 2-adic. —The preceding unsigned comment was added by 61.108.11.194 (talkcontribs) 06:41, 8 February 2007 (UTC)

That sounds correct, and it could be made more precise. For example, you could say that any summation method for divergent series that is both stable and linear, and sums 1 + 2 + 4 + 8 + · · ·, sums it to −1. Melchoir 22:17, 8 February 2007 (UTC)
A work colleague gave me a variant of this proof, and referred to it as “proof that the universe is two's complement”. (The two's complement representation of −1 is all-bits-one, which in some sense ought to mean 1+2+4+⋯.) Pjrm 06:28, 19 April 2007 (UTC)

Using the formula for the limiting sum anyway: = 1/(1-2) = -1

Looking at the series to the left of the term.. 1/2, 1/4.. etc we get limiting sum 1.

It's analogous to "running the integral" backwards of this exponential function and finding it's limiting area.

(remark)

I'd say its not 'n = 1 + 2n' but rather 'n = 1 + 2n-inf' —The preceding unsigned comment was added by Morgenrodeo (talkcontribs) 08:14, 28 March 2007 (UTC).

There is an error with the proof. you cannot multiply infinity by two since by definition it is the largest number possible. —The preceding unsigned comment was added by 172.207.30.115 (talkcontribs) 06:55, April 1, 2007 (UTC).

This isn't true. Infinity is not the "biggest number possible", it's not even a number. There are different types of infinity. For example the integers are infinite in number, and so are the real numbers, but there are far more real numbers than integers. The integers are countably infinite, the real numbers are uncountably infinite. The problem in the proof is that 1 + 2 + 4 + 8 + … is not well defined: it's infinite! It just doesn't make any sense to write 1 + 2 + 4 + 8 + … and expect it to mean anything. And so multiplying by 2 doesn't make any sense either. The formula for a geometric progression only works when | r | < 1. When | r | ≥ 1 then the sum diverges. The problem here is a common one; people are trying to apply a result without enforcing all of the prerequisite assumptions. ~~ Dr Dec (Talk) ~~ 22:48, 30 September 2009 (UTC)
Absolutely. The above "proof" multiplies one infinite quantity by two, and then subtracts another infinite quantity from it... and then expects a meaningful, finite answer. Most useful definitions of "infinity" in mathematics preclude that possibility.
Similar errors of subtracting infinite quantities lead to other nonsensical results. For example, here is a "proof" that 0=1, and that 0=2 also:
n = 1 + 1 + 1 + 1 + 1 + ...
n + 1 = 1 + 1 + 1 + 1 + 1 + ...
Thus n=n+1
Subtract n from both sides
Thus 0 = 1
n = 2 + 2 + 2 + 2 + 2 + ...
n + 2 = 2 + 2 + 2 + 2 + 2 + ...
Thus n=n+2
Thus 0 = 2
This mathematical fallacy is very much related to Hilbert's paradox of the Grand Hotel. Moxfyre (ǝɹʎℲxoɯ | contrib) 01:35, 2 October 2009 (UTC)

As presented in the original post this is not a fallacy; it is a perfectly valid proof that, as stated, if the series converges then it converges to -1. Since it is perfectly clear that it does not converge this result is of no conceivable value, but that is not at all the same as saying that it is fallacious. However, the section heading is nonsense: it is not a proof that the series converges to -1, and, in light of the statement from the original poster of what it proves, it is a bizarre heading. JamesBWatson (talk) 18:36, 4 October 2009 (UTC)

## Partial Sum

The line reading: Here n = 1+2+4+8+…….. 2n = 2+4+8+16+…………

Here we can easily say 2n>n

2n-n = (2+4+8+16+…………) – (1+2+4+8+……..) 2n-n = -1 n = 2n+1, which contradict 2n>n so n ≠ -1

## Don't merge individual series into here

The beauty of summary style is that we can develop examples in separate articles, which include unique applications and background. All convergent geometric series are not identical, and they can't be covered in a single article. Melchoir 18:47, 16 March 2007 (UTC)

Most of the content of the individual articles is already here — all convergent geometric series are (pretty much) identical except for where they're referred to in the literature. 1/2 + 1/4 + ... -> Zeno and 0.999, 9/10 + 9/100 + ... to 0.999, 1/2 - 1/4 + 1/8 - ... to Numbers, 1/4 + 1/16 + ... to the previous series (if it is notable), etc. Perhaps there should be an article should be examples of geometric series, split out from the "geometric series" section, here, but the ones here don't deserve an article because no one could type it it exactly, and search probably wouldn't fit it either. (The spaced "dots" rather than unspaced dots or an ellipsis is another problem.... (!)). — Arthur Rubin | (talk) 22:40, 16 March 2007 (UTC)
Well, the individual articles are mostly substubs created without my knowledge; that's why they currently lack unique content. You could make them into redirects, but then I'd just have to recreate them when I get around to it.
Anyway, this article has become an unreadable, tangled list of boring quasi-information. It's a concrete argument against the incremental improvement model. As a reader, if I followed a link to some series and it took me here, I'd feel betrayed. Melchoir 23:21, 16 March 2007 (UTC)

I am for the merge unless the article can be expanded into something unique enough to merit its own article--Cronholm144 22:55, 12 May 2007 (UTC)

## incorrect description of final outcome in one case

1. Positive, the terms will all be the same sign as the initial term.
2. Negative, the terms will alternate between positive and negative.
3. Greater than 1, there will be exponential growth towards positive infinity.

I think that the third line should read:

Greater than 1, there will be exponential growth towards infinity (of the same sign as the first term).

eg, -1, -2, -4, -8 .... common ratio is 2, and is > 1

This is only the second 'post' I have made to Wikipedia. I would rather put this as a discussion item than wade in and edit the main page.

If someone with more experience agrees, would they please alter the main page.

Thank you Jsryork 11:21, 7 May 2007 (UTC)

## What's the purpose of "a" in all these formulae?

Why is a needed in all these formulae? It's clear that it's a constant and therefore it can always come out of the summation... 83.67.217.254 20:41, 21 August 2007 (UTC)

It can come out of the summation when you're solving for the sum of the series, but it's needed when expressing the series as "a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number". If you didn't have a factor of a in each term, you could only express geometric series for which the first term is 1. —David Eppstein 05:09, 27 August 2007 (UTC)
That is not a property of the series. Its an application not an algebraic property. This is clutter. — Preceding unsigned comment added by 95.176.83.179 (talk) 10:37, 29 May 2011 (UTC)
Although not a significant improvement, it does allow the sum to be determined for an arbitrary geometric progression, not just one starting with "1", per David Eppstein's comment above. — Arthur Rubin (talk) 14:43, 29 May 2011 (UTC)

## Discrepancies in term numbering in formulae

"Elementary properties" says a_n = a *r^(n-1) i.e. terms start from n=1

however the summation formula and everything from the section "Geometric series" onwards uses: a_k = ar^k starting with k=0

2) The term-index used varies between n, k and even i

Please use one consistent notation and one term-index throughout; most people would favor the notation a_k = ar^k (i.e. numbering always starts from 0) and index k for simplicity. It also keeps the summation formula simpler.

Smcinerney 04:08, 3 September 2007 (UTC)

I have revised the Elementary Properties section to be consistent with the Series section. Journeyman (talk) 23:18, 26 March 2008 (UTC)

## Geometric series article

Hi, I just added an article entirely on the subject of infinite geometric series. Geometric progressions and geometric series are important enough that they deserve separate articles. Jim 21:06, 22 September 2007 (UTC)

## Formula for infinite series

${\displaystyle {\frac {1}{{\frac {1}{a^{1}}}+{\frac {1}{a^{2}}}+{\frac {1}{a^{3}}}+{\frac {1}{a^{4}}}+{\frac {1}{a^{5}}}\ldots }}={\frac {1}{\sum _{n=1}^{\infty }{\frac {1}{a^{n}}}}}=a-1;a>0}$

User:twentythreethousand 22:39, 27 September 2007

## Infinite Series Explained

I had an assignment involving infinite geometric series and at the time I couldn't understand the justification for the formula presented here, as it looked like half of the formula just dissapeared. Later I found that it was because the r^infinity equalled zero. I tried to simplify this line of math so that others wouldn't be as confused as I was... is it mathematically correct tho, ne comments Laubpatr (talk) 04:34, 16 March 2008 (UTC)

The argument you give is not quite correct, since r is not well-defined. However, it is true that if |r| < 1, then the value of rn can be made arbitrarily close to zero by selecting a sufficiently large value of n, that is, limn→∞ rn = 0. Michael Slone (talk) 08:23, 16 March 2008 (UTC)

## derivation and simple example

The formula ar^(n-1) is easily derived and simplifies understanding. As my edits are always deleted by some insane, wide eyed, fool of a moderater I sha'n't waste my time producing one. But a derivation would be a worthy addition to the page.

Also a simple example would make this more accessable. Perhaps using the forumula to calculate compound interest.

Also, where are the link to the exponential function and the derivation of e?

Page needs a lot of work. —Preceding unsigned comment added by 91.111.137.74 (talk) 13:49, 16 August 2009 (UTC)

## Error?

There appears to be an error in the geometric series section. The "convenient formula" appears to be wrong. —Preceding unsigned comment added by 67.230.157.187 (talk) 04:26, 1 October 2009 (UTC)

No,the formula is correct. We can derive it here. Let's ignore the factor of a, we could just multiply through by a at the and if we like. If Sn = 1 + r + r2 + … + rn then rSn = r + r2 + … + rn+1 and so (1 − r)S = 1 − rn+1. Dividing through by 1 − r gives
${\displaystyle S_{n}={\frac {1-r^{n+1}}{1-r}}={\frac {r^{n+1}-1}{r-1}}\ .}$
Maybe the sign change confused you? In the article the numerator and denominator have both been multiplied by −1. But since (−1) ÷ (−1) = +1 there is no change in the answer. ~~ Dr Dec (Talk) ~~ 10:34, 1 October 2009 (UTC)

P.S. When you post onto a talk page, please put your post at the bottom of the page. See the talk page article for more information. ~~ Dr Dec (Talk) ~~ 10:34, 1 October 2009 (UTC)

## start value? whazzat?

The introduction of the article currently says that the ratio between adjacent terms is "equal to the sequence's start value". That doesn't make sense to me, since I don't know what is meant by progression's start value. Perhaps the use of that phrase is a remnant from an earlier edition of the page wherein the constant "a" was not used?

In any case, can someone can clarify the phrase in the article, or remove it?

I know I'm supposed to Be Bold in editing, so maybe I should have removed the phrase, but this would have been my first edit to an article, and leading from ignorance was unappealing. Helpful suggestions are welcome. DavidHolmes0 (talk) 21:52, 6 September 2010 (UTC)

Please ignore my comments above, which, on later reading, don't seem to match the history of the article. What was I thinking? DavidHolmes0 (talk) 15:02, 21 September 2010 (UTC)

## Calculus on Geometric Series

Under the Geometric Series heading there is a line that says, "Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form." This is a tad bit confusing because it is first unclear which formula "this formula" is. Apart from simply editing for clarity, should this material be moved to a section about calculus? Thelema418 (talk) 05:25, 23 August 2012 (UTC)

## Merger proposal

I propose that Geometric progression be merged in part or whole into Geometric series. The concept of the progression or sequence is necessary to understand the series, so it is necessary in the Series article. For some reason, the series article concerns infinite series, but finite series are described in the geometric progression article. This is confusing because it makes it sound as though Geometric series refers to the infinite progression, and never a finite sequence. Thelema418 (talk) 02:20, 24 August 2012 (UTC)

I would agree; they should be together as are Arithmetic Progression and Arithmetic Series. It's confusing to have Geometric Progression and Geometric Series as separate, lengthy articles. Startswithj (talk) 18:48, 30 November 2013 (UTC)
I disagree with the merger proposal. See my comments on Talk:Geometric series#Merger proposal. Jim.belk (talk) 05:02, 26 January 2014 (UTC)
I also disagree. A progression and a serie are totally differents objects. As far as the merged proposal is concerned: both progression and series are infinite, which do not forbid to look at finite inital sub-series. 2A02:1205:C6B7:F9C0:8150:27F3:EB3E:54DC (talk) 17:19, 22 July 2014 (UTC)
Agree. See my comments in Talk:Geometric series#Merger proposal. Mario Castelán Castro (talk) 00:52, 8 February 2015 (UTC).
• Oppose merge. As the IPV6 editor said, series (adding things up) and progressions (listing things one after the other in a certain order) are two very different things, and blurring the distinctions between the two does no favors to our readers. Additionally, the geometric series article is already on the long side, and if the two are merged it will become more difficult to find information about geometric progressions within them. I do think that the "series" part of the progression article is too long — it should be cut back to a brief overview of the main results for geometric series with a pointer to the other article. —David Eppstein (talk) 02:12, 8 February 2015 (UTC)
• Oppose merge, Support two articles: I suggest that the articles Geometric progression and Geometric series be two different articles, as even the finite series stray from the topic. I would structure it as if something is related to the progression, it will be put into the progression, and if something is related to the series, it will be placed into the series. DSCrowned(talk) 06:32, 13 July 2016 (UTC)

## Finding R

Is it possible to find R given the sum, first term, and number of terms? I've asked my math teachers this and they all said no. I'm assuming that the responses here will be the same, but even if it's not possible, how close can I get to finding the answer? — Preceding unsigned comment added by 71.206.163.33 (talk) 21:22, 25 August 2013 (UTC)

In a sense, yes, if r is positive. S(a,r,n) is an increasing function of r for r nonnegative, so there is a unique solution of S(a,r,n)=S given a, n, and S if S > a. However, there isn't a "closed form" solution. — Arthur Rubin (talk) 01:03, 27 November 2013 (UTC)

## An error in product section

I notice that the product equal to P=(a0*a1)^((n+1)/2) The correct formula should be P=(a0*a1)^(n/2)

No, the formula given, ${\displaystyle P=(a_{0}a_{n})^{\frac {n+1}{2}}}$, is correct. There are n + 1 terms since the first term is indexed by "0", and the proof is straight-forward. Bill Cherowitzo (talk) 18:48, 24 April 2015 (UTC)