Una productoria es una forma abrevidada de escribir el producto de números; esta suma abreviada tiene una simbología propia, haciendo uso de un símbolo \(\Pi\), unos límites: inferior \(i=1\) inicializado por medio del contador \(i\), y un límite superior \(n\geq1\) un número entero; y finalmente un factores o multiplicando, valores a ser sumados \(x_i\in\mathbf{R}\) con \(i=1,2,\ldots,n\)
\[\prod_{i=1}^{n}x_i=x_1{\times}x_2{\times}\cdots{\times}x_n\]
Los factores pueden tomar cualquier valor, aunque se puede definir una regla que permita establecer que numeros quieren nultiplicarse.
Sean \(x_1=5\), \(x_2=3\), \(x_3=7\), \(x_4=1\) y \(x_5=-6\) y lo que se pretende es multiplicar estos valores
\[\prod_{i=1}^{5}x_i=5{\times}3{\times}7{\times}1{\times}(-6)\]
5*3*7*1*(-6)## [1] -630x <- c(5,3,7,1,-6)
prod(x)## [1] -630x[1]*x[2]*x[3]*x[4]*x[5]## [1] -630Dentro de la productoria se cumplen algunas propiedades, útiles a la hora de realizar productorias o entender como funcionan en sí mismas. Se destacaran solo algunas:
\[\prod_{i=1}^{n}k=k^{n};\ con\ n\in\textbf{Z}^{+}\ y \ k\ una\ constante\]
\[\prod_{i=1}^{5}4={4}^{5}=4\times4\times4\times4\times4=1024\]
4*4*4*4*4## [1] 1024y <- c(4, 4, 4, 4, 4)
y## [1] 4 4 4 4 4prod(y)## [1] 1024\[\prod_{i=1}^{23}4=4^{23}=7,036874{\times}{10}^{13}=70,36874{\times}{10}^{12}=703,6874{\times}{10}^{11}=70.368.740.000.000\]
z <- rep(x=4, times=23)
z## [1] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4prod(z)## [1] 7.036874e+13prod(rep(x = 4, times =23))## [1] 7.036874e+13\[\prod_{i=1}^{n}i=n!=1{\times}2{\times}{\cdots}{\times}(n-1){\times}n=n{\times}(n-1){\times}{\cdots}{\times}2{\times}1;\ con\ n\in\textbf{Z}^{+}\]
\[\prod_{i=1}^{5}i=5!=1{\times}2{\times}3{\times}4{\times}5=5{\times}4{\times}3{\times}2{\times}1=120\]
w <- 1:5
prod(w)## [1] 120factorial(5)## [1] 120\[\prod_{i=1}^{10}i=10!=10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1=3.628.800\]
v <- 1:10
prod(v)## [1] 3628800factorial(10)## [1] 3628800\[\prod_{i=1}^{50}i=50!=50{\times}49{\times}48{\times}{\cdots}{\times}3{\times}2{\times}1=3,041409{\times}{10}^{64}=30,41409{\times}{10}^{63}=304,1409{\times}{10}^{62}=3041,409{\times}{10}^{61}\]
prod(1:50)## [1] 3.041409e+64factorial(50)## [1] 3.041409e+64\[\prod_{i=1}^{n}\left(i+k\right)=\frac{\left(n+k\right)!}{k!};\ con\ n\in\textbf{Z}^{+}\ y \ k\ una\ constante\]
\[\prod_{i=1}^{4}\left(i+3\right)=\frac{\left(4+3\right)!}{3!}=\frac{7!}{3!}=\frac{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{3{\times}2{\times}1}=7{\times}6{\times}5{\times}4=840\]
h <- 7:4
prod(h)## [1] 840h <- 4:7
prod(h)## [1] 840factorial(4+3)/factorial(3)## [1] 840\[\prod_{i=1}^{7}\left(i+2\right)=\frac{\left(7+2\right)!}{2!}=\frac{9!}{2!}=\frac{9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{2{\times}1}=9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3=181.440\]
i <- 9:3
prod(i)## [1] 181440factorial(7+2)/factorial(2)## [1] 181440\[\prod_{i=1}^{6}\left(i+4\right)=\frac{\left(6+4\right)!}{4!}=\frac{10!}{4!}=\frac{10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{4{\times}3{\times}2{\times}1}=10{\times}9{\times}8{\times}7{\times}6{\times}5=151.200\]
j <- 10:5
prod(j)## [1] 151200factorial(6+4)/factorial(4)## [1] 151200\[\prod_{i=m}^{n}i=\frac{n!}{\left(m-1\right)!};\ con\ m,\ n\in\textbf{Z}^{+}\ y\ m{\leq}n\]
\[\prod_{i=4}^{10}i=\frac{10!}{\left(4-1\right)!}=\frac{10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{3!}=\frac{10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{3{\times}2{\times}1}=10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4=604.800\]
k <- 10:4
prod(k)## [1] 604800factorial(10)/factorial(4-1)## [1] 604800\[\prod_{i=6}^{10}i=\frac{10!}{\left(6-1\right)!}=\frac{10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{5!}=\frac{10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{5{\times}4{\times}3{\times}2{\times}1}=10{\times}9{\times}8{\times}7{\times}6=30.240\]
k <- 10:6
prod(k)## [1] 30240factorial(10)/factorial(6-1)## [1] 30240\[\prod_{i=8}^{12}i=\frac{12!}{\left(8-1\right)!}=\frac{12{\times}11{\times}10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{7!}=\frac{12{\times}11{\times}10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}=12{\times}11{\times}10{\times}9{\times}8=95.040\]
k <- 12:8
prod(k)## [1] 95040factorial(12)/factorial(8-1)## [1] 95040\[\prod_{i=1}^{n}x_{i}y_{i}=\prod_{i=1}^{n}x_{i}\prod_{i=1}^{n}y_{i};\ con\ x_{i},y_{i}\in\textbf{R},\ n\in\textbf{Z}^{+}\]
\[\prod_{i=1}^{5}{3\left(i+3\right)}=\prod_{i=1}^{5}{3}\prod_{i=1}^{5}{\left(i+3\right)}={3}^{5}\frac{\left(5+3\right)!}{3!}=243\frac{8!}{3{\times}2{\times}1}=243\frac{8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{3{\times}2{\times}1}=243\cdot8{\times}7{\times}6{\times}5{\times}4=243\cdot6.720=1.632.960\]
e <- rep(x=3,times=5)
f <- (8:4)
prod(e)*prod(f)## [1] 16329603**5*factorial(5+3)/factorial(3)## [1] 1632960\[\prod_{i=1}^{5}{i}{\left(i+3\right)}=\prod_{i=1}^{5}{i}\prod_{i=1}^{5}{\left(i+3\right)}={5!}\frac{\left(5+3\right)!}{3!}=5{\times}4{\times}3{\times}2{\times}1\frac{8!}{3{\times}2{\times}1}=5{\times}4\cdot8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1=20\cdot40.320=806.400\]
g <- (5:4)
h <- (8:1)
prod(g)*prod(h)## [1] 806400factorial(5)*factorial(5+3)/factorial(3)## [1] 806400\[\prod_{i=1}^{n}\frac{x_{i}}{y_{i}}=\frac{\prod\limits_{i=1}^{n}x_{i}}{\prod\limits_{i=1}^{n}y_{i}};\ con\ x_{i},y_{i}\in\textbf{R},y_{i}\neq0\ y\ n\in\textbf{Z}^{+}\]
\[\prod_{i=1}^{7}\frac{i+3}{i}=\frac{\prod\limits_{i=1}^{7}{(i+3)}}{\prod\limits_{i=1}^{7}{i}}=\frac{\frac{\left(7+3\right)!}{3!}}{{7!}}=\frac{\frac{10!}{3{\times}2{\times}1}}{{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}}=\frac{\frac{10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{3{\times}2{\times}1}}{\frac{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{1}}=\frac{10{\times}9{\times}8{\times}7{\times}6{\times}5{\times}4}{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}=\frac{10{\times}9{\times}8}{3{\times}2{\times}1}=120\]
i <- (10:8)
j <- (3:1)
prod(i)/prod(j)## [1] 120(factorial(7+3)/factorial(3))/factorial(7)## [1] 120\[\prod_{i=1}^{7}\frac{i+2}{i}=\frac{\prod\limits_{i=1}^{7}{(i+2)}}{\prod\limits_{i=1}^{7}{i}}=\frac{\frac{\left(7+2\right)!}{2!}}{{7!}}=\frac{\frac{9!}{2{\times}1}}{{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}}=\frac{\frac{9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{2{\times}1}}{\frac{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{1}}=\frac{9{\times}8{\times}7{\times}6{\times}5{\times}4{\times}3}{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}=\frac{9{\times}8}{2{\times}1}=36\]
k <- (9:8)
l <- (2:1)
prod(k)/prod(l)## [1] 36(factorial(7+2)/factorial(2))/factorial(7)## [1] 36\[\prod_{i=1}^{n}\frac{x_{i+1}}{x_{i}}=\frac{x_{n+1}}{x_{1}};\ con\ x_{i}\in\textbf{R},x_{i}\neq0\ y\ n\in\textbf{Z}^{+}\]
\[\prod_{i=1}^{7}\frac{i+1}{i}=\frac{\prod\limits_{i=1}^{7}{(i+1)}}{\prod\limits_{i=1}^{7}{i}}=\frac{\frac{\left(7+1\right)!}{1!}}{{7!}}=\frac{\frac{8!}{1}}{{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}}=\frac{\frac{8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{1}}{\frac{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{1}}=\frac{8{\times}7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}{7{\times}6{\times}5{\times}4{\times}3{\times}2{\times}1}=8\]
m <- (8:1)
n <- (7:1)
prod(m)/prod(n)## [1] 8(factorial(7+1)/factorial(1))/factorial(7)## [1] 8(7+1)/1## [1] 8\[\prod_{i=1}^{70}\frac{i+1}{i}=\frac{\prod\limits_{i=1}^{70}{i+1}}{\prod\limits_{i=1}^{70}{i}}=\frac{\frac{\left(70+1\right)!}{1!}}{{70!}}=\frac{\frac{71!}{1}}{{70{\times}69{\times}\cdots{\times}2{\times}1}}=\frac{\frac{71{\times}70{\times}\cdots{\times}2{\times}1}{1}}{\frac{70{\times}69{\times}\cdots{\times}2{\times}1}{1}}=\frac{71{\times}70{\times}\cdots{\times}2{\times}1}{70{\times}69{\times}\cdots{\times}2{\times}1}=71\]
o <- (71:1)
p <- (70:1)
prod(o)/prod(p)## [1] 71(factorial(70+1)/factorial(1))/factorial(70)## [1] 71(70+1)/1## [1] 71\[\prod_{i=1}^{7000}\frac{i+1}{i}=\frac{\prod\limits_{i=1}^{7000}{i+1}}{\prod\limits_{i=1}^{7000}{i}}=\frac{\frac{\left(7000+1\right)!}{1!}}{{7000!}}=\frac{\frac{7001!}{1}}{{7000{\times}6999{\times}\cdots{\times}2{\times}1}}=\frac{\frac{7001{\times}7000{\times}\cdots{\times}2{\times}1}{1}}{\frac{7000{\times}6999{\times}\cdots{\times}2{\times}1}{1}}=\frac{7001{\times}7000{\times}\cdots{\times}2{\times}1}{7000{\times}6999{\times}\cdots{\times}2{\times}1}=7001\]
q <- (7001:1)
r <- (7000:1)
prod(q)/prod(r)## [1] NaN(factorial(7000+1)/factorial(1))/factorial(7000)## [1] NaN(7000+1)/1## [1] 7001Sean \(x_1=2\), \(x_2=3\), \(x_3=7\), \(x_4=1\) y \(x_5=-6\)
\[\prod_{i=1}^{4}\frac{x_{i+1}}{x_{i}}=\frac{x_{4+1}}{x_{1}}=\frac{x_{5}}{x_{1}}=\frac{-6}{2}=-3\]
\[\prod_{i=1}^{4}\frac{x_{i+1}}{x_{i}}=\frac{x_{2}}{x_{1}}\cdot\frac{x_{3}}{x_{2}}\cdot\frac{x_{4}}{x_{3}}\cdot\frac{x_{5}}{x_{4}}=\frac{3}{2}\cdot\frac{7}{3}\cdot\frac{1}{7}\cdot\frac{-6}{1} =\frac{-6}{2}=-3\]
s <- c(2,3,7,1,-6)prod(s[2:5])/prod(s[1:4])## [1] -3s[5]/s[1]## [1] -3\[\prod_{i=1}^{n}kx_{i}=k^{n}\prod_{i=1}^{n}x_{i};\ con\ x_{i}\in\textbf{R},\ n\in\textbf{Z}^{+}\ y\ k\ una\ constante\]
\[\prod_{i=1}^{12}2{i}=2^{12}\prod_{i=1}^{12}{i}=2^{12}{\times}12!=4096{\times}12!=4.096{\times}479.001.600=1,961991{\times}10^{12}=19,61991{\times}10^{11}=196,1991{\times}10^{10}=1.961,991{\times}10^{9}\]
t <- rep(x=2, times=12)
u <- 1:12
prod(t)*prod(u)## [1] 1.961991e+122**12*factorial(12)## [1] 1.961991e+12\[\prod_{i=1}^{7}2{\left(i+4\right)}=2^{7}\prod_{i=1}^{7}{\left(i+4\right)}=2^{7}{\cdot}\frac{\left(7+4\right)!}{4!}=128{\cdot}\frac{11!}{4!}=128{\times}11{\times}10{\times}9{\times}8{\times}7{\times}6{\times}5=128{\times}1.663.200=212.889.600\]
v <- rep(x=2, times=7)
w <- 11:5
prod(v)*prod(w)## [1] 212889600x <- 1:7+4
prod(v)*prod(x)## [1] 2128896002**7*factorial(7+4)/factorial(4)## [1] 212889600