63
$\begingroup$

(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?

4 Answers 4

88

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

  • 0
    Guess I was too tired to see that, thanks.2017-10-12
33

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 $

  • 0
    @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

  • 0
    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
4

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