Difference between revisions of "Elementary symmetric sum"

m
Line 1: Line 1:
 
== Definition ==
 
== Definition ==
  
The ''n''th symmetric sum is the sum of a group of numbers, taken ''n'' at a timeSo if the numbers are a, b, c, and d, then:
+
The $k$-th '''symmetric sum''' of a [[set]] of $n$ numbers is the sum of all products of $k$ of those numbers ($1 \leq k \leq n)For example, if $n = 4$, and our set of numbers is $\{a, b, c, d\}$, then:
  
1st Symmetric Sum = a+b+c+d
+
1st Symmetric Sum = $a+b+c+d$
  
2nd Symmetric Sum = ab+ac+ad+bc+bd+cd
+
2nd Symmetric Sum = $ab+ac+ad+bc+bd+cd$
  
3rd Symmetric Sum = abc+abd+acd+bcd
+
3rd Symmetric Sum = $abc+abd+acd+bcd$
  
4th Symmetric Sum = abcd
+
4th Symmetric Sum = $abcd$
  
  

Revision as of 13:12, 24 July 2006

Definition

The $k$-th symmetric sum of a set of $n$ numbers is the sum of all products of $k$ of those numbers ($1 \leq k \leq n). For example, if $n = 4$, and our set of numbers is $\{a, b, c, d\}$, then:

1st Symmetric Sum = $a+b+c+d$

2nd Symmetric Sum = $ab+ac+ad+bc+bd+cd$

3rd Symmetric Sum = $abc+abd+acd+bcd$

4th Symmetric Sum = $abcd$


Uses

Symmetric sums show up in Vieta's formulas

Invalid username
Login to AoPS