Difference between revisions of "Summation"

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 $\left(\sum \right)$ .

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

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

Look for 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 $m + n.$ (1997 AIME, #2)

@@ Line 1: / Line 1: @@
-A '''summation''' is a form of shorthand used foten.
+A '''summation''' is the [[sum]] of a number of terms (addends). Summations are often written using sigma notation <math>\left(\sum \right)</math>.
-==Definitions==
+==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>.
+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==
+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=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^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>
-==Special Summations==
+=== Intermediate ===
-Certain types of summations are different from the common variety, such as [[cyclic sum]]s, and [[symmetric sum]]s.
+*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==
 *[[Cyclic sum]]
 *[[Symmetric sum]]
+*[[PaperMath’s sum]]
 [[Category:Definition]]

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