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> | ||
+ | |||
+ | *<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> | ||
+ | Look for PaperMath’s sums if you want to look deeper into these identities | ||
+ | |||
+ | == 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== | ||
*[[Cyclic sum]] | *[[Cyclic sum]] | ||
*[[Symmetric sum]] | *[[Symmetric sum]] | ||
− | + | *[[PaperMath’s sum]] | |
[[Category:Definition]] | [[Category:Definition]] |
Latest revision as of 16:39, 8 October 2023
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
Or
Look for PaperMath’s sums if you want to look deeper into these identities
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)