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(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)

I'm aware of Sigma notation, but is there a function/name for e.g.

$$ 4 + 3 + 2 + 1 \longrightarrow 10 ,$$

similar to $$4! = 4 \cdot 3 \cdot 2 \cdot 1 ,$$ which uses multiplication?

Edit: I found what I was looking for, but is there a name for this type of summation?

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    See [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber's_formula).2017-01-12
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    I like to call it "additorial" or "sumitorial" :)2018-04-23
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    @AlexM. absolutely not. He's asking for a term, not a proof of an equality.2018-10-31
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    @TheGreatDuck: That post contains the required term, too. Reading it also gives an answer to this question. Keep in mind that on StackExchange sites "duplicate" does not mean "exact duplicate", but rather "loose duplicate".2018-10-31
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    @AlexM. That's a stupid reason to mark as duplicate though. When something is marked as duplicate the test says "marked as **exact** duplicate". Has that been revised since I last saw it?2018-10-31
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    @TheGreatDuck: Needlessly calling people "stupid" is not the best approach in life. Anyway, the problem of whether to interpret the word "duplicate" in a strict or loose sense has caused a lot of discussion, proof that it is not a simple one. For a start, go patiently through the answers given [here](https://math.meta.stackexchange.com/q/23458) and [here](https://math.meta.stackexchange.com/q/18917). I also believe that battling the closure of some random post is not the most helpful thing to do on MSE.2018-11-01

4 Answers 4

84

The name for

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

is the $n$th triangular number. This picture demonstrates the reasoning for the name:

$$T_1=1\qquad T_2=3\qquad T_3=6\qquad T_4=10\qquad T_5=15\qquad T_6=21$$

$\hskip1.7in$ enter image description here

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    What does the last notation in brackets mean? Does it have a name?2016-03-29
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    @silkfire - It's called a [binomial coefficient](https://en.wikipedia.org/wiki/Binomial_coefficient)2016-04-04
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    @RussellThackston Thanks!2016-04-04
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    How could we adapt this to be used with a known base number ? Let's say we have a base value of 7. Then we need 7+14, the we need 7+14+21 and so forth. Could this be turned into an excel formula ?2017-10-12
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    @Overmind: Note that $$7+14+21+\cdots+(7\times n)=7\times(1+2+3+\cdots+n)=7\times T_n$$ So the formula is quite simple :)2017-10-12
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    Guess I was too tired to see that, thanks.2017-10-12
32

Donald Knuth in The Art of Computer Programming calls the $n$-th triangular number the "termial function", and denotes it

$$n? = 1 + 2 + ... + n = \sum_{k=1}^n k $$

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    Wow, really? What volume/page does he define this termin-ology?2011-08-30
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    Volume 1, section 1.2.5 I believe, it is in the "Permutations and Factorials" section.2011-08-30
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    Now that I am home, I have the 3rd edition of volume one, it is on page 48.2011-08-30
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    I just looked it up. He describes it precisely for the purpose of contrasting with the factorial function, and the name seems to be a play on words (term-inal rather than factor-ial). I was suspicious that he would give such prominence to such an elementary bit of mathematics, but it makes sense in the name of pedagogy.2011-08-30
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    It's not _terminal_, it's _termial_. It also doesn't matter why he put it in his books, it is exactly what the questioner was asking about.2011-08-30
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    ah, then I managed to misread it several times. Also –– if you will forgive me –– I was somewhat skeptical that Knuth would deign to give this function a name (especially when I thought that name was supposed to be "terminal", which made little sense to me); I wanted to see for myself, and also see why he would do so.2011-08-30
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    @Niel: concerning "for pedagical reasons": I'd say, the additive analogon of a "factor" in a multiplication is "summand", so then it should rather be called "summorial" or "summatorial"2011-10-04
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    @GottfriedHelms term is also used to refer to a summand, and it as a much better ring then summatorial which is a mouthful. Termial rolls of the tongue which while that is a minor note, it is important that terms being coined in any field aren't difficult to say. It's part of the art of naming.2018-10-31
8

Actually, I've found what I was looking for.

From the wiki on Summation:

enter image description here

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    These numbers are also called the triangular numbers. You might think of the triangular numbers as naming a sequence: 1, 3, 6, 10, 15, 21,... But a sequence of integers is really just a function from $\mathbb{N}$ to $\mathbb{Z}$, so the triangular numbers also name the function you've written above.2011-08-29
3

Not exactly a name, but note that $$ \sum\limits_{k=1}^{n} k= \frac{n(n+1)}{2}={n+1 \choose 2} $$