Difference between revisions of "Cyclic sum"
m (→Rigorous definition: spelling) |
m (→Rigorous definition: closing parentheses) |
||
| Line 2: | Line 2: | ||
==Rigorous definition== | ==Rigorous definition== | ||
| − | Consider a function <math>f(a_1,a_2,a_3,\ldots a_n</math>. The cyclic sum <math>\sum f(a_1,a_2,a_3,\ldots a_n)</math> is equal to | + | Consider a function <math>f(a_1,a_2,a_3,\ldots a_n)</math>. The cyclic sum <math>\sum f(a_1,a_2,a_3,\ldots a_n)</math> is equal to |
<cmath>f(a_1,a_2,a_3,\ldots a_n)+f(a_2,a_3,a_4,\ldots a_n,a_1)+f(a_3,a_4,\ldots a_n,a_1,a_2)\ldots+f(a_n,a_1,a_2,\ldots a_{n-1})</cmath> | <cmath>f(a_1,a_2,a_3,\ldots a_n)+f(a_2,a_3,a_4,\ldots a_n,a_1)+f(a_3,a_4,\ldots a_n,a_1,a_2)\ldots+f(a_n,a_1,a_2,\ldots a_{n-1})</cmath> | ||
Revision as of 11:04, 23 November 2007
A cyclic sum is a summation that cycles through all the values of a function and takes their sum, so to speak.
Rigorous definition
Consider a function 
. The cyclic sum 
is equal to
![\[f(a_1,a_2,a_3,\ldots a_n)+f(a_2,a_3,a_4,\ldots a_n,a_1)+f(a_3,a_4,\ldots a_n,a_1,a_2)\ldots+f(a_n,a_1,a_2,\ldots a_{n-1})\]](http://latex.artofproblemsolving.com/9/4/f/94f2282a9eaced21157c0de211a1267cabb983f1.png)
Note that not all permutations of the variables are used; they are just cycled through.
Notation
If a summation is specified without additional arguments, it is generally assumed to be a cyclic sum. A cyclic sum can also be specified by having the variables to cycle through underneath the sigma, as follows: 
. Note that a cyclic sum need not cycle through all of the variables.
