Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Summation"

m (→‎Definition)
m (Undo revision 241755 by Maxd3 (talk) Formatting, remove vandalism)
(Tag: Undo)
 
(4 intermediate revisions by 3 users not shown)
Line 1: Line 1:
 
A '''summation''' is the [[sum]] of a number of terms (addends). Summations are often written using sigma notation <math>\left(\sum \right)</math>.
 
A '''summation''' is the [[sum]] of a number of terms (addends). Summations are often written using sigma notation <math>\left(\sum \right)</math>.
  
−
==Definition==
+
== 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>. 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.
+
 
 +
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>.  
 
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>.  
  
−
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>.
+
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==
+
== 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}(f(i)+g(i))=\sum_{i=a}^{b}f(i)+\sum_{i=a}^{b}g(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=a}^{b}c\cdot f(i)=c\cdot \sum_{i=a}^{b}f(i)</math>
−
*<math>\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}</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^3 = \left(\sum_{i=1}^{n} i\right)^2 = \left(\frac{n(n+1)}{2}\right)^2</math>
+
* <math>\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}</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=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} {n\choose i} = 2^n</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,j}^{n} = \sum_i^n \sum_j^n</math>
+
* <math>\sum_{i=0}^{n} {n\choose i} = 2^n</math>
 +
* <math>\sum_{i,j}^{n} = \sum_i^n \sum_j^n</math>
 +
* <math>\sum_{i=0}^{2n} {(x^2 \times 10^i)}=(\sum_{j=0}^n {(3x \times 10^j)})^2 + \sum_{k=0}^n {(2x^2 \times 10^k)}</math>
 +
Or
 +
* <math>x^2\sum_{i=0}^{2n} {10^i}=(x \sum_{j=0}^n {(3 \times 10^j)})^2 + x^2\sum_{k=0}^n {(2 \times 10^k)}</math>
 +
See [[PaperMath’s sum]]s if you want to look deeper into these identities
  
 
== Problems ==
 
== Problems ==
 +
 
=== Introductory ===
 
=== Introductory ===
−
*Evaluate the following sums:
+
 
−
**<math>\sum_{i=1}^{20} i</math>
+
* Evaluate the following sums:
−
**<math>\sum_{i=5}^{15} i + 1</math>
+
** <math>\sum_{i=1}^{20} i</math>
−
**<math>\sum_{i=1}^{9} {10\choose i}</math>
+
** <math>\sum_{i=5}^{15} i + 1</math>
 +
** <math>\sum_{i=1}^{9} {10\choose i}</math>
  
 
=== Intermediate ===
 
=== 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]])
+
*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|Source]])
  
 
=== Olympiad ===
 
=== Olympiad ===
Line 35: Line 45:
  
 
[[Category:Definition]]
 
[[Category:Definition]]
 +
 +
{{stub}}

Latest revision as of 14:11, 2 February 2025

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 $\sum_{i=a}^{b} c^i = \frac{c^{b+1}-c^a}{c-1}$
  • $\sum_{i=0}^{n} {n\choose i} = 2^n$
  • $\sum_{i,j}^{n} = \sum_i^n \sum_j^n$
  • $\sum_{i=0}^{2n} {(x^2 \times 10^i)}=(\sum_{j=0}^n {(3x \times 10^j)})^2 + \sum_{k=0}^n {(2x^2 \times 10^k)}$

Or

  • $x^2\sum_{i=0}^{2n} {10^i}=(x \sum_{j=0}^n {(3 \times 10^j)})^2 + x^2\sum_{k=0}^n {(2 \times 10^k)}$

See PaperMath’s sums if you want to look deeper into these identities

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 $m + n.$

(Source)

Olympiad

See Also

This article is a stub. Help us out by expanding it.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Summation&oldid=241777"