Difference between revisions of "Summation"

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A '''summation''' is a form of shorthand used often.
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A '''summation''' is the [[sum]] of a number of terms (addends). Summations are often written using sigma notation <math>\left(\sum \right)</math>.
  
==Definitions==
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==Definition==
For <math>b\ge a</math>, and a set <math>c</math> (or any other algebraic structure), <math>\sum_{i=a}^{b}c_i=c_a+c_{a+1}+c_{a+2}...+c_{b-1}+c_{b}</math>. Note that if <math>a>b</math>, then the sum is <math>0</math>.
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For <math>b\ge a</math>, and a set <math>c</math> (or any other algebraic structure), <math>\sum_{i=a}^{b}c_i=c_a+c_{a+1}+c_{a+2}...+c_{b-1}+c_{b}</math>. Here <math>i</math> refers to the index of summation, <math>a</math> is the lower bound, and <math>b</math> is the upper bound.
  
==Rules==
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As an example, <math>\sum_{i=3}^6 i^3 = 3^3 + 4^3 + 5^3 + 6^3</math>. Note that if <math>a>b</math>, then the sum is <math>0</math>.
*<math>\sum_{i=a}^{b}f(i)+g(i)=\sum_{i=a}^{b}f(i)+\sum_{i=a}^{b}g(i)</math>
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Quite often, sigma notation is used in a slightly different format to denote certain sums. For example, <math>\sum_{cyc}</math> refers to a [[cyclic sum]], and <math>\sum_{a,b \in S}</math> refers to all subsets <math>a, b</math> which are in <math>S</math>. Usually, the bottom of the sigma contains a logical condition, as in <math>\sum_{i|n}^{n} i</math>, where the sum only includes the terms <math>i</math> which divide into <math>n</math>.
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==Identities==
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*<math>\sum_{i=a}^{b}(f(i)+g(i))=\sum_{i=a}^{b}f(i)+\sum_{i=a}^{b}g(i)</math>
 
*<math>\sum_{i=a}^{b}c\cdot f(i)=c\cdot \sum_{i=a}^{b}f(i)</math>
 
*<math>\sum_{i=a}^{b}c\cdot f(i)=c\cdot \sum_{i=a}^{b}f(i)</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>
 
*<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^2 = \frac{n(n+1)(2n+1)}{6}</math>
 
*<math>\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}</math>
 
*<math>\sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2 = \left(\frac{n(n+1)}{2}\right)^2</math>
 
*<math>\sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2 = \left(\frac{n(n+1)}{2}\right)^2</math>
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*<math>\sum_{i=0}^{n} x^n = \frac{x^{n+1}-1}{x-1}</math>, and in general <math>\sum_{i=a}^{b} c^i = \frac{c^{b+1}-c^a}{c-1}</math>
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*<math>\sum_{i=0}^{n} {n\choose i} = 2^n</math>
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*<math>\sum_{i,j}^{n} = \sum_i^n \sum_j^n</math>
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== Problems ==
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=== Introductory ===
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*Evaluate the following sums:
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**<math>\sum_{i=1}^{20} i</math>
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**<math>\sum_{i=5}^{15} i + 1</math>
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**<math>\sum_{i=1}^{9} {10\choose i}</math>
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=== Intermediate ===
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*The nine horizontal and nine vertical lines on an <math>8\times8</math> checkerboard form <math>r</math> [[rectangles]], of which <math>s</math> are [[square]]s.  The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math> ([[1997 AIME Problems/Problem 2|1997 AIME, #2]])
  
==Special Summations==
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=== Olympiad ===
Certain types of summations are different from the common variety, such as [[cyclic sum]]s, and [[symmetric sum]]s.
 
  
 
==See Also==
 
==See Also==

Latest revision as of 22:21, 14 March 2022

A summation is the sum of a number of terms (addends). Summations are often written using sigma notation $\left(\sum \right)$.

Definition

For $b\ge a$, and a set $c$ (or any other algebraic structure), $\sum_{i=a}^{b}c_i=c_a+c_{a+1}+c_{a+2}...+c_{b-1}+c_{b}$. Here $i$ refers to the index of summation, $a$ is the lower bound, and $b$ is the upper bound.

As an example, $\sum_{i=3}^6 i^3 = 3^3 + 4^3 + 5^3 + 6^3$. Note that if $a>b$, then the sum is $0$.

Quite often, sigma notation is used in a slightly different format to denote certain sums. For example, $\sum_{cyc}$ refers to a cyclic sum, and $\sum_{a,b \in S}$ refers to all subsets $a, b$ which are in $S$. Usually, the bottom of the sigma contains a logical condition, as in $\sum_{i|n}^{n} i$, where the sum only includes the terms $i$ which divide into $n$.

Identities

  • $\sum_{i=a}^{b}(f(i)+g(i))=\sum_{i=a}^{b}f(i)+\sum_{i=a}^{b}g(i)$
  • $\sum_{i=a}^{b}c\cdot f(i)=c\cdot \sum_{i=a}^{b}f(i)$
  • $\sum_{i=1}^{n} i= \frac{n(n+1)}{2}$, and in general $\sum_{i=a}^{b} i= \frac{(b-a+1)(a+b)}{2}$
  • $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$
  • $\sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2 = \left(\frac{n(n+1)}{2}\right)^2$
  • $\sum_{i=0}^{n} x^n = \frac{x^{n+1}-1}{x-1}$, and in general $\sum_{i=a}^{b} c^i = \frac{c^{b+1}-c^a}{c-1}$
  • $\sum_{i=0}^{n} {n\choose i} = 2^n$
  • $\sum_{i,j}^{n} = \sum_i^n \sum_j^n$

Problems

Introductory

  • Evaluate the following sums:
    • $\sum_{i=1}^{20} i$
    • $\sum_{i=5}^{15} i + 1$
    • $\sum_{i=1}^{9} {10\choose i}$

Intermediate

  • The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ (1997 AIME, #2)

Olympiad

See Also

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