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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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4 Answers 4

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. Here's a picture that 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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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? –  Niel de Beaudrap Aug 30 '11 at 1:06
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. –  Niel de Beaudrap Aug 30 '11 at 10:41
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. –  tlehman Aug 30 '11 at 12:38
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. –  Niel de Beaudrap Aug 30 '11 at 12:41
@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" –  Gottfried Helms Oct 4 '11 at 6:14

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. –  Jonas Kibelbek Aug 29 '11 at 20:59

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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