# What is the term for a factorial type operation, but with summation instead of products?

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

From the wiki on Summation:

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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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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:

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

$\hskip1.7in$

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Donald Knuth in The Art of Computer Programming calls the $n$-th triangular number the "termial function", and denotes it
$$n? = 1 + 2 + ... + n = \sum_{k=1}^n k$$