Difference between revisions of "User:Temperal/The Problem Solver's Resource3"

(Rules of Summation: improve)
(Rules of Summation: note to self)
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<math>\sum_{i=1}^{n} i= \frac{n(n+1)}{2}</math>, and in general <math>\sum_{i=a}^{b} i= \frac{(b-a+1)(a+b)}{2}</math>
 
<math>\sum_{i=1}^{n} i= \frac{n(n+1)}{2}</math>, and in general <math>\sum_{i=a}^{b} i= \frac{(b-a+1)(a+b)}{2}</math>
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The above should all be self-evident and provable by the reader within seconds.
  
 
<math>\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}</math>
 
<math>\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}</math>
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<math> \sum_{i=1}^n i^5=\frac{n^2(n+1)^2(2n^2+2n-1)}{12}</math>
 
<math> \sum_{i=1}^n i^5=\frac{n^2(n+1)^2(2n^2+2n-1)}{12}</math>
  
<!--there are others, I forgot what they were. Could someone please fill them in? -->
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<!-- eliminate useless n=4 and n=5; add proof for n=2 which easily generalizes to general n -->
  
 
===Rules of Products===
 
===Rules of Products===

Revision as of 22:52, 10 January 2009


Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 3.

Summations and Products

Definitions

  • Summations: $\sum_{i=a}^{b}c_i=c_a+c_{a+1}+c_{a+2}...+c_{b-1}+c_{b}$
  • Products: $\prod_{i=a}^{b}c_i=c_a\cdot c_{a+1}\cdot c_{a+2}...\cdot c_{b-1}\cdot c_{b}$

Rules of Summation

$\sum_{i=a}^{b}f_1(i)+f_2(i)+\ldots f_n(i)=\sum_{i=a}^{b}f(i)+\sum_{i=a}^{b}f_2(i)+\ldots+\sum_{i=a}^{b}f_n(i)$

$\sum_{i=a}^{b}c\cdot f(i)=c\cdot \sum_{i=a}^{b}f(i)$

$\sum_{i=1}^{n} i= \frac{n(n+1)}{2}$, and in general $\sum_{i=a}^{b} i= \frac{(b-a+1)(a+b)}{2}$

The above should all be self-evident and provable by the reader within seconds.

$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

$\sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2 = \left(\frac{n(n+1)}{2}\right)^2$

$\sum_{i=1}^n i^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$

$\sum_{i=1}^n i^5=\frac{n^2(n+1)^2(2n^2+2n-1)}{12}$


Rules of Products

$\prod_{i=a}^{b}x=x^{(b-a+1)}$

$\prod_{i=a}^{b}x\cdot y=x^{(b-a+1)}y^{(b-a+1)}$


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