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This question already has an answer here:

(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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marked as duplicate by Alex M., Namaste algebra-precalculus Oct 31 '18 at 11:02

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ See Faulhaber's formula. $\endgroup$ – Lucian Jan 12 '17 at 11:25
  • $\begingroup$ I like to call it "additorial" or "sumitorial" :) $\endgroup$ – NH. Apr 23 '18 at 14:52
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    $\begingroup$ @AlexM. absolutely not. He's asking for a term, not a proof of an equality. $\endgroup$ – The Great Duck Oct 31 '18 at 3:56
  • $\begingroup$ @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". $\endgroup$ – Alex M. Oct 31 '18 at 9:07
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    $\begingroup$ @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? $\endgroup$ – The Great Duck Oct 31 '18 at 23:16
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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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    $\begingroup$ What does the last notation in brackets mean? Does it have a name? $\endgroup$ – silkfire Mar 29 '16 at 22:50
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    $\begingroup$ @silkfire - It's called a binomial coefficient $\endgroup$ – Russell Thackston Apr 4 '16 at 17:33
  • $\begingroup$ @RussellThackston Thanks! $\endgroup$ – silkfire Apr 4 '16 at 19:37
  • $\begingroup$ 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 ? $\endgroup$ – Overmind Oct 12 '17 at 6:45
  • $\begingroup$ @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 :) $\endgroup$ – Zev Chonoles Oct 12 '17 at 7:39
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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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    $\begingroup$ Wow, really? What volume/page does he define this termin-ology? $\endgroup$ – Niel de Beaudrap Aug 30 '11 at 1:06
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    $\begingroup$ Now that I am home, I have the 3rd edition of volume one, it is on page 48. $\endgroup$ – tlehman Aug 30 '11 at 2:14
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    $\begingroup$ 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. $\endgroup$ – tlehman Aug 30 '11 at 12:38
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    $\begingroup$ 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. $\endgroup$ – Niel de Beaudrap Aug 30 '11 at 12:41
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    $\begingroup$ @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" $\endgroup$ – Gottfried Helms Oct 4 '11 at 6:14
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Actually, I've found what I was looking for.

From the wiki on Summation:

enter image description here

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  • $\begingroup$ 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. $\endgroup$ – Jonas Kibelbek Aug 29 '11 at 20:59
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Not exactly a name, but note that $$ \sum\limits_{k=1}^{n} k= \frac{n(n+1)}{2}={n+1 \choose 2} $$

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