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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

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