Difference between revisions of "Cyclic sum"
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==Rigorous definition== | ==Rigorous definition== | ||
| − | Consider a function <math>f(a_1,a_2,a_3,\ldots a_n)</math>. The cyclic sum <math>\sum_{cyc} 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_{cyc} 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, a_5, \ldots, a_n, a_1, a_2) + \ldots + f(a_n, a_1, a_2, \ldots, a_{n-1}).</cmath> |
Note that not all permutations of the variables are used; they are just cycled through. | Note that not all permutations of the variables are used; they are just cycled through. | ||
Revision as of 08:09, 28 April 2022
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
![\[f(a_1, a_2, a_3, \ldots, a_n) + f(a_2, a_3, a_4, \ldots, a_n, a_1) + f(a_3, a_4, a_5, \ldots, a_n, a_1, a_2) + \ldots + f(a_n, a_1, a_2, \ldots, a_{n-1}).\]](http://latex.artofproblemsolving.com/d/e/9/de9cd5ba1ed1eee9688ac0937d1aba4515d4b126.png)
Note that not all permutations of the variables are used; they are just cycled through.
Notation
A cyclic sum is often 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.
A cyclic sum is also sometimes specified by 
. This notation implies that all variables are cycled through.
