{"id":1422,"date":"2010-10-26T11:28:21","date_gmt":"2010-10-26T09:28:21","guid":{"rendered":"http:\/\/laaventuradelasmatematicas.jesussoto.es\/?p=1422"},"modified":"2013-01-27T15:27:09","modified_gmt":"2013-01-27T15:27:09","slug":"el-binomio-de-newton-por-induccion","status":"publish","type":"post","link":"https:\/\/pimedios.jesussoto.es\/?p=1422","title":{"rendered":"El binomio de Newton por inducci\u00f3n"},"content":{"rendered":"<p>Tengo una especial predilecci\u00f3n por las matem\u00e1ticas de los siglos XVII y XVIII, por ello vamos a continuar con la demostraci\u00f3n de un cl\u00e1sico: el binomio de Newton. Esta vez demostrado por inducci\u00f3n.<\/p>\n<p>Consideremos la proposici\u00f3n matem\u00e1tica<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/binomiNewton1.png\" width=\"212\" height=\"60\" \/><\/p>\n<p>y veamos que ocurre para <em>n=1<\/em>:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/binomiNewton2.png\" width=\"400\" height=\"61\" \/><\/p>\n<p>La proposici\u00f3n se cumple. Ahora supongamos que se cumple para cierto <em>n=k<\/em>; es decir,<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/binomiNewton3.png\" width=\"209\" height=\"53\" \/><\/p>\n<p style=\"text-align: left;\">Veamos que tambi\u00e9n se cumple para <em>n=k+1<\/em>; es decir,<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/binomiNewton4.png\" width=\"272\" height=\"53\" \/><\/p>\n<p style=\"text-align: left;\">Para demostrarlo, podemos calcular $ (a+b)^{k+1}=(a+b)(a+b)^k$ y aplicar la hip\u00f3tesis de inducci\u00f3n para $ (a+b)^k$. Por tanto,<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/binomiNewton5.png\" width=\"400\" height=\"506\" \/><\/p>\n<p style=\"text-align: left;\">Con lo que queda demostrado para todo <em>n <\/em>entero positivo.<\/p>\n<h3>Referencia:<\/h3>\n<ul>\n<li><a href=\"http:\/\/books.google.es\/books?id=t_7o7YN-3cUC&amp;lpg=PA463&amp;dq=induccion%20matematica&amp;pg=PA463#v=onepage&amp;q&amp;f=false\">Matem\u00e1ticas Fundamentales Para Ingenieros<\/a>, google books<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostraci\u00f3n del binomio de Newton por inducci\u00f3n<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[268],"class_list":["post-1422","post","type-post","status-publish","format-standard","hentry","category-ocio","tag-newton","entry"],"_links":{"self":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/1422","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1422"}],"version-history":[{"count":3,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/1422\/revisions"}],"predecessor-version":[{"id":3651,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/1422\/revisions\/3651"}],"wp:attachment":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1422"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1422"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1422"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}