# Difference between revisions of "Summation"

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− | A '''summation''' is a | + | A '''summation''' is the [[sum]] of a number of terms (addends). Summations are often written using sigma notation <math>\left(\sum \right)</math>. |

− | == | + | ==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>. | + | 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. |

− | == | + | 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> | + | |

+ | 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>. | ||

+ | |||

+ | ==Identities== | ||

+ | *<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> | ||

+ | *<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> | ||

+ | *<math>\sum_{i=0}^{n} {n\choose i} = 2^n</math> | ||

+ | *<math>\sum_{i,j}^{n} = \sum_i^n \sum_j^n</math> | ||

+ | |||

+ | == Problems == | ||

+ | === Introductory === | ||

+ | *Evaluate the following sums: | ||

+ | **<math>\sum_{i=1}^{20} i</math> | ||

+ | **<math>\sum_{i=5}^{15} i + 1</math> | ||

+ | **<math>\sum_{i=1}^{9} {10\choose i}</math> | ||

+ | |||

+ | === Intermediate === | ||

+ | *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]]) | ||

− | == | + | === Olympiad === |

− | |||

==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 .

## Contents

## Definition

For , and a set (or any other algebraic structure), . Here refers to the index of summation, is the lower bound, and is the upper bound.

As an example, . Note that if , then the sum is .

Quite often, sigma notation is used in a slightly different format to denote certain sums. For example, refers to a cyclic sum, and refers to all subsets which are in . Usually, the bottom of the sigma contains a logical condition, as in , where the sum only includes the terms which divide into .

## Identities

- , and in general
- , and in general

## Problems

### Introductory

- Evaluate the following sums:

### Intermediate

- The nine horizontal and nine vertical lines on an checkerboard form rectangles, of which are squares. The number can be written in the form where and are relatively prime positive integers. Find (1997 AIME, #2)