Difference between revisions of "Elementary symmetric sum"

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== Definition ==
 
== 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:
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The <math>k</math>-th '''symmetric sum''' of a [[set]] of <math>n</math> numbers is the sum of all products of <math>k</math> of those numbers (<math>1 \leq k \leq n</math>).  For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
  
1st Symmetric Sum = $a+b+c+d$
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1st Symmetric Sum = <math>a+b+c+d</math>
  
2nd Symmetric Sum = $ab+ac+ad+bc+bd+cd$
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2nd Symmetric Sum = <math>ab+ac+ad+bc+bd+cd</math>
 
 
3rd Symmetric Sum = $abc+abd+acd+bcd$
 
 
 
4th Symmetric Sum = $abcd$
 
  
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3rd Symmetric Sum = <math>abc+abd+acd+bcd</math>
  
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4th Symmetric Sum = <math>abcd</math>
  
 
== Uses ==
 
== Uses ==
  
 
Symmetric sums show up in [[Vieta's formulas]]
 
Symmetric sums show up in [[Vieta's formulas]]

Revision as of 17:14, 26 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

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