# Cyclic sum

 This is an AoPSWiki Word of the Week for February 4-11

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 $f(a_1,a_2,a_3,\ldots a_n)$. The cyclic sum $\sum f(a_1,a_2,a_3,\ldots a_n)$ is equal to

$$f(a_1,a_2,a_3,\ldots a_n)+f(a_2,a_3,a_4,\ldots a_n,a_1)+f(a_3,a_4,\ldots a_n,a_1,a_2)\ldots+f(a_n,a_1,a_2,\ldots a_{n-1})$$

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: $\sum_{a,b,c}\frac{ab}{cd}$. Note that a cyclic sum need not cycle through all of the variables.