Associated Topics || Dr. Math Home || Search Dr. Math

### Sum of First n Natural Numbers

```Date: 12/03/2005 at 01:35:34
From: Kevin
Subject: the additive analog of a factorial

The product of a series of consecutive integers is called a factorial.
This seems to be a standard term and even has a standard symbol (!).
It's defined that n! = 1*2*3*...*n.  Is there a similar term for the
sum of consecutive integers (1+2+3+...+n)?  If so, what is it, and
does it have a standard symbol?

```

```
Date: 12/03/2005 at 23:22:34
From: Doctor Peterson
Subject: Re: the additive analog of a factorial

Hi, Kevin.

A lot of people ask us this.  Some ask if there is a "summorial", or
something like that!

The sum of the first n natural numbers is called the nth triangular
number.  This is because it can be seen as the number of dots in a
triangle like this, the 5th triangular number:

o
o o
o o o    = 1+2+3+4+5 = 15
o o o o
o o o o o

There is no real symbol for it, in the sense of the factorial symbol,
probably because it is easily calculated, as n(n+1)/2 (unlike the
factorial, which is very hard to get without doing the
multiplications).  But the nth triangular number is often written as
something like

T
n

with the n as a subscript.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search