{"id":377,"date":"2024-06-10T12:31:37","date_gmt":"2024-06-10T10:31:37","guid":{"rendered":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/?p=377"},"modified":"2024-07-24T11:13:39","modified_gmt":"2024-07-24T09:13:39","slug":"el-triangulo-de-tartaglia-o-de-pascal-y-su-aplicacion","status":"publish","type":"post","link":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/el-triangulo-de-tartaglia-o-de-pascal-y-su-aplicacion\/","title":{"rendered":"El tri\u00e1ngulo de Tartaglia o de Pascal y su aplicaci\u00f3n"},"content":{"rendered":"\n

El tri\u00e1ngulo de Tartaglia o de Pascal es harto conocido en el mundo de las matem\u00e1ticas ya desde tiempos pret\u00e9ritos. Su forma es la siguiente:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Este tri\u00e1ngulo es enormemente \u00fatil para establecer los coeficientes en ecuaciones binomiales elevados a la n-\u00e9sima potencia, es decir:<\/p>\n\n\n

~~~ (x + y)^n = c_0 x^{n-0} y^0 + c_1 x^{n-1} y^1 + c_2 x^{n-2} y^2 + ... + c_{n-1} x^{n-(n-1)} y^{n-1} + c_n x^{n-n} y^0 <\/span>
\n ~~~ (x + y)^n = c_0 x^n + c_1 x^{n-1} y + c_2 x^{n-2} y^2 + ... + c_{n-1} x y^{n-1} + c_n y^0 <\/span><\/p>\n

Los coeficientes c_i <\/span> que aparecen en el desarrollo de las potencias binomiales son los que se extraen del tri\u00e1ngulo de Tartaglia.<\/p>\n\n\n\n

Vayamos a ejemplos concretos y con ello verificaremos la utilidad de esta construcci\u00f3n matem\u00e1tica.<\/p>\n\n\n

~~~ n=0 ~~~ \\longrightarrow ~~~ (x + y)^0 = 1 <\/span>
\n ~~~ n=1 ~~~ \\longrightarrow ~~~ (x + y)^1 = x^1 + y^1 <\/span>
\n ~~~ n=2 ~~~ \\longrightarrow ~~~ (x + y)^2 = x^2 + 2 x y + y^2 <\/span>
\n ~~~ n=3 ~~~ \\longrightarrow ~~~ (x + y)^3 = x^3 + 3 x^2 y + 3 x y^2 + y^3 <\/span>
\n ~~~ n=4 ~~~ \\longrightarrow ~~~ (x + y)^4 = x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4 <\/span>
\n ~~~ n=5 ~~~ \\longrightarrow ~~~ (x + y)^5 = x^5 + 5 x^4 y + 10 x^3 y^2 + 10 x^2 y^3 + 5 x y^4 + y^5 <\/span>
\n ~~~ n=6 ~~~ \\longrightarrow ~~~ (x + y)^6 = x^6 + 6 x^5 y + 15 x^4 y^2 + 20 x^3 y^3 + 15 x^2 y^4 + 6 x y^5 + y^6 <\/span>
\n ~~~ n=7 ~~~ \\longrightarrow ~~~ (x + y)^7 = x^7 + 7 x^6 y + 21 x^5 y^2 + 35 x^4 y^3 + 35 x^3 y^4 + 21 x^2 y^5 + 7 x y^6 + y^7 <\/span>
\n ~~~ n=8 ~~~ \\longrightarrow ~~~ (x + y)^8 = x^8 + 8 x^7 y + 28 x^6 y^2 + 56 x^5 y^3 + 70 x^4 y^4 + 56 x^3 y^5 + 28 x^2 y^6 + 8 x y^7 + y^8 <\/span>
\n ~~~ .................................................. <\/span><\/p>\n\n\n\n

Este tri\u00e1ngulo se utilizar\u00e1 tambi\u00e9n en combinatoria mediante la regla de Pascal, donde generaremos un tri\u00e1ngulo tal que:<\/p>\n\n\n

\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\\begin{pmatrix}\n\n0 \\\\\n\n0\n\n\\end{pmatrix}\n\n<\/span>
\n\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n\\begin{pmatrix}\n\n1 \\\\\n\n0\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n1 \\\\\n\n1\n\n\\end{pmatrix}\n\n<\/span>
\n\n\n~~~~~~~~~~~~~~~~~~~~~~~~\n\n\\begin{pmatrix}\n\n2 \\\\\n\n0\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n2 \\\\\n\n1\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n2 \\\\\n\n2\n\n\\end{pmatrix}\n\n<\/span>
\n\n\n~~~~~~~~~~~~~~~~\n\n\\begin{pmatrix}\n\n3 \\\\\n\n0\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n3 \\\\\n\n1\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n3 \\\\\n\n2\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n3 \\\\\n\n3\n\n\\end{pmatrix}\n\n<\/span>
\n\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n4 \\\\\n\n0\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n4 \\\\\n\n1\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n4 \\\\\n\n2\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n4 \\\\\n\n3\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n4 \\\\\n\n4\n\n\\end{pmatrix}\n\n<\/span>
\n\n\n\\begin{pmatrix}\n\n5 \\\\\n\n0\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n5 \\\\\n\n1\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n5 \\\\\n\n2\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n5 \\\\\n\n3\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n5 \\\\\n\n4\n\n\\end{pmatrix}\n\n~~~~~~~~\n\n\\begin{pmatrix}\n\n5 \\\\\n\n5\n\n\\end{pmatrix}\n\n<\/span>
\n\n\n..................................................................................\n\n<\/span><\/p>\n\n\n\n

De forma que podr\u00edamos expresar:<\/p>\n\n\n ~~~ (x + y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k <\/span>\n\n\n\n

sabiendo que:<\/p>\n\n\n

~~~ \\binom{n}{k} = \\frac{n!}{k! (n-k)!} <\/span>
\n~<\/span>
\n ~~~ n! = n~\u00b7~(n-1)~\u00b7~(n-2)~\u00b7~...~\u00b7~2~\u00b7~1 <\/span><\/p>\n\n\n\n

Veamos el desarrollo con un ejemplo con n=3:<\/p>\n\n\n

~~~ n=3<\/span>
\n ~~~~~~~~~~~~~~~ (x + y)^3 = \\binom{3}{0} x^3 + \\binom{3}{1} x^2 y + \\binom{3}{2} x y^2 + \\binom{3}{3} y^3 <\/span>
\n~<\/span>
\n ~~~~~~~~~~~~~~~ (x + y)^3 = \\frac{3!}{0! 3!} x^3 + \\frac{3!}{1! 2!} x^2 y + \\frac{3!}{2! 1!} x y^2 + \\frac{3!}{3! 0!} y^3 <\/span>
\n~<\/span>
\n ~~~~~~~~~~~~~~~ (x + y)^3 = \\frac{3~\u00b7~2~\u00b7~1}{1~\u00b7~3~\u00b7~2~\u00b7~1} x^3 + \\frac{3~\u00b7~2~\u00b7~1}{1~\u00b7~2~\u00b7~1} x^2 y + \\frac{3~\u00b7~2~\u00b7~1}{2~\u00b7~1~\u00b7~1} x y^2 + \\frac{3~\u00b7~2~\u00b7~1}{3~\u00b7~2~\u00b7~1~\u00b7~1} y^3 <\/span>
\n~<\/span>
\n ~~~~~~~~~~~~~~~ (x + y)^3 = \\frac{6}{6} x^3 + \\frac{6}{2} x^2 y + \\frac{6}{2} x y^2 + \\frac{6}{6} y^3 <\/span>
\n~<\/span>
\n ~~~~~~~~~~~~~~~ (x + y)^3 = x^3 + 3 x^2 y + 3 x y^2 + y^3 <\/span><\/p>\n\n\n\n

Una de la propiedades m\u00e1s sorprendentes es la que relaciona este tri\u00e1ngulo con la serie de Fibonacci<\/a> y la proporci\u00f3n \u00e1urea, y sino observad los datos:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

El tri\u00e1ngulo de Tartaglia es enormemente \u00fatil para establecer los coeficientes en ecuaciones binomiales elevados a la n-\u00e9sima potencia.<\/p>\n","protected":false},"author":3,"featured_media":532,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_seopress_robots_primary_cat":"none","_seopress_titles_title":"%%post_title%%","_seopress_titles_desc":"%%post_excerpt%%","_seopress_robots_index":"","footnotes":""},"categories":[16],"tags":[50,51,9,47,49,48],"class_list":["post-377","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra","tag-binomial","tag-ecuaciones-binomiales","tag-serie-de-fibonacci","tag-tartaglia","tag-triangulo-de-pascal","tag-triangulo-de-tartaglia"],"_links":{"self":[{"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/posts\/377","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/comments?post=377"}],"version-history":[{"count":119,"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/posts\/377\/revisions"}],"predecessor-version":[{"id":1009,"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/posts\/377\/revisions\/1009"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/media\/532"}],"wp:attachment":[{"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/media?parent=377"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/categories?post=377"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/zynetyka.com\/la-espiral-de-la-vida\/wp-json\/wp\/v2\/tags?post=377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}