§1.2 Elementary Algebra

Permalink:: http://dlmf.nist.gov/1.2

§1.2(i) Binomial Coefficients
§1.2(ii) Finite Series
§1.2(iii) Partial Fractions
§1.2(iv) Means

§1.2(i) Binomial Coefficients

Notes:: See Chrystal (1959, pp. 62–70).
Keywords:: binomials
Referenced by:: §26.3(iv), Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/1.2.i
Errata (effective with 1.0.5):: The condition for (1.2.2), (1.2.4), and (1.2.5) was corrected. These equations are true only if $n$ is a positive integer. Previously $n$ was allowed to be zero.
Reported 2011-08-10 by Michael Somos

In (1.2.1) and (1.2.3) $k$ and $n$ are nonnegative integers and $k\leq n$ . In (1.2.2), (1.2.4), and (1.2.5) $n$ is a positive integer.

1.2.1 $\binom{n}{k}=\frac{n!}{(n-k)!k!}=\binom{n}{n-k}.$

Defines:: $\binom{m}{n}$ : binomial coefficient
Symbols:: $!$ : $n!$ : factorial, $k$ : integer, $m$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 3.1.2
Referenced by:: §1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E1
Encodings:: TeX, pMML, png

¶ Binomial Theorem

Keywords:: binomial theorem

1.2.2 $(a+b)^{n}=a^{n}+\binom{n}{1}a^{{n-1}}b+\binom{n}{2}a^{{n-2}}b^{2}+\dots+\binom{n}{n-1}ab^{{n-1}}+b^{n}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient and $n$ : nonnegative integer
A&S Ref:: 3.1.1
Referenced by:: §1.2(i), §1.2(i), Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/1.2.E2
Encodings:: TeX, pMML, png

1.2.3 $\binom{n}{0}+\binom{n}{1}+\dots+\binom{n}{n}=2^{n}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient and $n$ : nonnegative integer
A&S Ref:: 3.1.6
Referenced by:: §1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E3
Encodings:: TeX, pMML, png

1.2.4 $\binom{n}{0}-\binom{n}{1}+\dots+(-1)^{n}\binom{n}{n}=0.$

Symbols:: $\binom{m}{n}$ : binomial coefficient and $n$ : nonnegative integer
A&S Ref:: 3.1.7
Referenced by:: §1.2(i), §1.2(i), Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/1.2.E4
Encodings:: TeX, pMML, png

1.2.5 $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\dots+\binom{n}{\ell}=2^{{n-1}},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\ell$ : integer and $n$ : nonnegative integer
Referenced by:: §1.2(i), §1.2(i), Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/1.2.E5
Encodings:: TeX, pMML, png

where $\ell$ is $n$ or $n-1$ according as $n$ is even or odd.

In (1.2.6)–(1.2.9) $k$ and $m$ are nonnegative integers and $n$ is unrestricted.

1.2.6 $\binom{n}{k}=\frac{n(n-1)\cdots(n-k+1)}{k!}=\frac{(-1)^{k}\left(-n\right)_{{k}}}{k!}=(-1)^{k}\binom{k-n-1}{k}.$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $k$ : integer and $n$ : nonnegative integer
A&S Ref:: 3.1.3
Referenced by:: ¶ ‣ §1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E6
Encodings:: TeX, pMML, png

1.2.7 $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : nonnegative integer
A&S Ref:: 3.1.4
Permalink:: http://dlmf.nist.gov/1.2.E7
Encodings:: TeX, pMML, png

1.2.8 $\sum^{m}_{{k=0}}\binom{n+k}{k}=\binom{n+m+1}{m}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $k$ : integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.2.E8
Encodings:: TeX, pMML, png

1.2.9 $\binom{n}{0}-\binom{n}{1}+\dots+(-1)^{m}\binom{n}{m}=(-1)^{m}\binom{n-1}{m}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $m$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §1.2(i)
Permalink:: http://dlmf.nist.gov/1.2.E9
Encodings:: TeX, pMML, png

§1.2(ii) Finite Series

Notes:: See Chrystal (1959, pp. 482–483, 489).
Permalink:: http://dlmf.nist.gov/1.2.ii

¶ Arithmetic Progression

Keywords:: arithmetic progression

1.2.10 $a+(a+d)+(a+2d)+\dots+(a+(n-1)d)=na+\tfrac{1}{2}n(n-1)d=\tfrac{1}{2}n(a+\ell),$

Symbols:: $n$ : nonnegative integer
A&S Ref:: 3.1.9
Permalink:: http://dlmf.nist.gov/1.2.E10
Encodings:: TeX, pMML, png

where $\ell$ = last term of the series = $a+(n-1)d$ .

¶ Geometric Progression

Keywords:: geometric progression (or series)

1.2.11 $a+ax+ax^{2}+\dots+ax^{{n-1}}=\frac{a(1-x^{n})}{1-x},$ $x\not=1$ .

Symbols:: $n$ : nonnegative integer
A&S Ref:: 3.1.10
Permalink:: http://dlmf.nist.gov/1.2.E11
Encodings:: TeX, pMML, png

§1.2(iii) Partial Fractions

Keywords:: partial fractions
Referenced by:: ¶ ‣ §5.19(i)
Permalink:: http://dlmf.nist.gov/1.2.iii

Let $\alpha _{1},\alpha _{2},\dots,\alpha _{n}$ be distinct constants, and $f(x)$ be a polynomial of degree less than $n$ . Then

1.2.12 $\frac{f(x)}{(x-\alpha _{1})(x-\alpha _{2})\cdots(x-\alpha _{n})}=\frac{A_{1}}{x-\alpha _{1}}+\frac{A_{2}}{x-\alpha _{2}}+\dots+\frac{A_{n}}{x-\alpha _{n}},$

Symbols:: $n$ : nonnegative integer and $A_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/1.2.E12
Encodings:: TeX, pMML, png

where

1.2.13 $A_{j}=\frac{f(\alpha _{j})}{\prod\limits _{{k\not=j}}(\alpha _{j}-\alpha _{k})}.$

Symbols:: $j$ : integer, $k$ : integer and $A_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/1.2.E13
Encodings:: TeX, pMML, png

Also,

1.2.14 $\frac{f(x)}{(x-\alpha _{1})^{n}}=\frac{B_{1}}{x-\alpha _{1}}+\frac{B_{2}}{(x-\alpha _{1})^{2}}+\dots+\frac{B_{n}}{(x-\alpha _{1})^{n}},$

Symbols:: $n$ : nonnegative integer and $B_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/1.2.E14
Encodings:: TeX, pMML, png

where

1.2.15 $B_{j}=\frac{f^{{(n-j)}}(\alpha _{1})}{(n-j)!},$

Symbols:: $!$ : $n!$ : factorial, $j$ : integer, $n$ : nonnegative integer and $B_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/1.2.E15
Encodings:: TeX, pMML, png

and $f^{{(k)}}$ is the $k$ -th derivative of $f$ (§1.4(iii)).

If $m_{1},m_{2},\dots,m_{n}$ are positive integers and $\deg f<\sum _{{j=1}}^{n}m_{j}$ , then there exist polynomials $f_{j}(x)$ , $\deg f_{j}<m_{j}$ , such that

1.2.16 $\frac{f(x)}{(x-\alpha _{1})^{{m_{1}}}(x-\alpha _{2})^{{m_{2}}}\cdots(x-\alpha _{n})^{{m_{n}}}}=\frac{f_{1}(x)}{(x-\alpha _{1})^{{m_{1}}}}+\frac{f_{2}(x)}{(x-\alpha _{2})^{{m_{2}}}}+\cdots+\frac{f_{n}(x)}{(x-\alpha _{n})^{{m_{n}}}}.$

Symbols:: $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.2.E16
Encodings:: TeX, pMML, png

To find the polynomials $f_{j}(x)$ , $j=1,2,\dots,n$ , multiply both sides by the denominator of the left-hand side and equate coefficients. See Chrystal (1959, pp. 151–159).

§1.2(iv) Means

Notes:: See Hardy et al. (1967, pp. 12–15).
Keywords:: arithmetic mean, geometric mean, harmonic mean, means, weighted means
Referenced by:: §1.7(iii)
Permalink:: http://dlmf.nist.gov/1.2.iv

The arithmetic mean of $n$ numbers $a_{1},a_{2},\dots,a_{n}$ is

1.2.17 $A=\frac{a_{1}+a_{2}+\dots+a_{n}}{n}.$

Defines:: $A$ : arithmetic mean
Symbols:: $n$ : nonnegative integer
A&S Ref:: 3.1.11
Permalink:: http://dlmf.nist.gov/1.2.E17
Encodings:: TeX, pMML, png

The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_{1},a_{2},\dots,a_{n}$ are given by

1.2.18 $G=(a_{1}a_{2}\cdots a_{n})^{{1/n}},$

Defines:: $G$ : geometric mean
Symbols:: $n$ : nonnegative integer
A&S Ref:: 3.1.12
Permalink:: http://dlmf.nist.gov/1.2.E18
Encodings:: TeX, pMML, png

1.2.19 $\frac{1}{H}=\frac{1}{n}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\dots+\frac{1}{a_{n}}\right).$

Defines:: $H$ : harmonic mean
Symbols:: $n$ : nonnegative integer
A&S Ref:: 3.1.13
Permalink:: http://dlmf.nist.gov/1.2.E19
Encodings:: TeX, pMML, png

If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_{1},a_{2},\dots,a_{n}$ , and $n$ positive numbers $p_{1},p_{2},\dots,p_{n}$ with