# What is $(N)+ (N-1) + (N -2) + \cdots + 1$ called?

This is purely for figuring the name of a mathematical concept.

For example, $$N \times (N-1) \times \cdots \times 1$$ is called factorial.

Question: What is $$N + (N-1) + (N -2) + \cdots + 1$$ called?

• Is it a Sequence? May 12, 2019 at 21:24
• @AndreasRejbrand I am wondering what is the name of this thing? May 12, 2019 at 21:25

The sum of numbers from $$1$$ to $$n$$ is called a "Triangular number".

From Wikipedia:

The triangle numbers are given by the following explicit formulas: $$T_n = \sum_{k=1}^n{k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}} = \binom{n + 1}{2},$$

So, the first triangular numbers are:

$$T_1 = 1\\ T_2 = 1 + 2 = 3\\ T_3 = 1 + 2 + 3 = 6\\ T_4 = 1 + 2 + 3 + 4 = 10\\ ...etc$$

• why did n(n+1)/2 become (n+1)/2 ? Shouldn't it be (n^2 + n)/2 ?
– TSR
Mar 16, 2023 at 1:43
• @TSR The second part isn't $(n + 1) / 2$, it's the Binomial coefficient, or the Combination, it is read as "$n + 1$ choose $2$" and computes the number of possible combinations of size $2$ from a set of size $n + 1$. Mar 16, 2023 at 21:01