{"id":32778,"date":"2019-11-08T16:54:14","date_gmt":"2019-11-08T19:54:14","guid":{"rendered":"https:\/\/blog.estrategiavestibulares.com.br\/?p=32778"},"modified":"2023-06-15T18:00:56","modified_gmt":"2023-06-15T21:00:56","slug":"principio-da-inducao-finita","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/principio-da-inducao-finita\/","title":{"rendered":"Princ\u00edpio da Indu\u00e7\u00e3o Finita"},"content":{"rendered":"<p>O princ&iacute;pio da indu&ccedil;&atilde;o finita ou simplesmente indu&ccedil;&atilde;o &eacute; um m&eacute;todo <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/category\/matematica\/\" target=\"_blank\">matem&aacute;tico<\/a> que pode ser utilizado para demonstrar proposi&ccedil;&otilde;es feitas no dom&iacute;nio dos n&uacute;meros naturais. O m&eacute;todo consiste em seguir os seguintes passos, para uma dada proposi&ccedil;&atilde;o:<div class=\"wp-block-buttons is-horizontal is-content-justification-center is-layout-flex wp-container-core-buttons-is-layout-1 wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-luminous-vivid-amber-to-luminous-vivid-orange-gradient-background has-background wp-element-button\" href=\"https:\/\/www.estrategiavestibulares.com.br\/vestibulares\/\" style=\"border-radius:28px\" target=\"_blank\">CURSOS PARA O VESTIBULAR<\/a><\/div>\n<\/div><ol class=\"block-editor-rich-text__editable editor-rich-text__editable\" role=\"textbox\" contenteditable=\"true\" aria-multiline=\"true\" data-is-placeholder-visible=\"false\" aria-label=\"Escrever lista...\" aria-autocomplete=\"list\">\n<li>Passo inicial: provar que a proposi&ccedil;&atilde;o &eacute; v&aacute;lida para o menor valor de n.<\/li>\n<li>Passo indutivo: assumido que a proposi&ccedil;&atilde;o &eacute; v&aacute;lida para um certo <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B150%7D&amp;space;%5Clarge&amp;space;k%5Cin&amp;space;%5Cmathbb%7BN%7D\" alt=\"\\dpi{150} \\large k\\in \\mathbb{N}\" align=\"absmiddle\"> e provar, a partir disso, que a proposi&ccedil;&atilde;o seria v&aacute;lida para o pr&oacute;ximo termo, ou seja, o termo <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B150%7D&amp;space;%5Clarge&amp;space;k+1\" alt=\"\\dpi{150} \\large k+1\" align=\"absmiddle\">.<\/li>\n<\/ol><p>Vejamos um exemplo para entender melhor:<\/p><h2><strong>Exemplo do Princ&iacute;pio de Indu&ccedil;&atilde;o Finita<\/strong><\/h2><p>Demonstre que <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B150%7D&amp;space;%5Clarge&amp;space;1+2+...+n=%5Cfrac%7Bn(n+1)%7D%7B2%7D\" alt=\"\\dpi{150} \\large 1+2+...+n=\\frac{n(n+1)}{2}\" align=\"absmiddle\"><\/p><p><strong>Passo Inicial:<\/strong><\/p><p>Vamos testar para <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B150%7D&amp;space;%5Clarge&amp;space;n=1\" alt=\"\\dpi{150} \\large n=1\" align=\"absmiddle\">:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B150%7D&amp;space;%5Clarge&amp;space;1=%5Cfrac%7B1(1+1)%7D%7B2%7D\" alt=\"\\dpi{150} \\large 1=\\frac{1(1+1)}{2}\" align=\"absmiddle\">&nbsp; (ok)<\/p><p><strong>Passo indutivo:<\/strong><\/p><p>Suponha que, para um certo k:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B150%7D&amp;space;%5Clarge&amp;space;1+2+...+k=%5Cfrac%7Bk(k+1)%7D%7B2%7D\" alt=\"\\dpi{150} \\large 1+2+...+k=\\frac{k(k+1)}{2}\" align=\"absmiddle\"><\/p><p>Vamos analisar a soma <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B150%7D&amp;space;%5Clarge&amp;space;1+2+...+k+(k+1)\" alt=\"\\dpi{150} \\large 1+2+...+k+(k+1)\" align=\"absmiddle\">.<\/p><p>Utilizando a hip&oacute;tese de indu&ccedil;&atilde;o finita, temos que<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;1+2+...+k+(k+1)=%5Cfrac%7Bk(k+1)%7D%7B2%7D+(k+1)=%5Cfrac%7Bk(k+1)%7D%7B2%7D+%5Cfrac%7B2(k+1)%7D%7B2%7D=%5Cfrac%7B(k+1)(k+2)%7D%7B2%7D\" alt=\"\\dpi{100} \\large 1+2+...+k+(k+1)=\\frac{k(k+1)}{2}+(k+1)=\\frac{k(k+1)}{2}+\\frac{2(k+1)}{2}=\\frac{(k+1)(k+2)}{2}\" align=\"absmiddle\"><\/p><p><strong>Desse modo:<\/strong><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;1+2+...+k+k+1=%5Cfrac%7B(k+1)(k+2)%7D%7B2%7D=%5Cfrac%7B(k+1)%5B(k+1)+1%5D%7D%7B2%7D\" alt=\"\\dpi{100} \\large 1+2+...+k+k+1=\\frac{(k+1)(k+2)}{2}=\\frac{(k+1)[(k+1)+1]}{2}\" align=\"absmiddle\">, o que prova nossa hip&oacute;tese de indu&ccedil;&atilde;o e tamb&eacute;m a proposi&ccedil;&atilde;o.<\/p><p>Esse m&eacute;todo &eacute; v&aacute;lido para provar uma proposi&ccedil;&atilde;o pois o passo indutivo prova que se o enunciado &eacute; v&aacute;lido para um n&uacute;mero qualquer, ent&atilde;o &eacute; v&aacute;lido para o pr&oacute;ximo. A isso, associa-se o passo inicial, que provou que a proposi&ccedil;&atilde;o &eacute; v&aacute;lida para o menor <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/historia-dos-numeros-complexos\/\" target=\"_blank\">n&uacute;mero<\/a> poss&iacute;vel. Desse modo, &eacute; poss&iacute;vel chegar em qualquer outro n&uacute;mero a partir da&iacute;.<\/p><figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.estrategiavestibulares.com.br\/\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" width=\"1024\" height=\"256\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2-1024x256.jpg\" alt=\"\" class=\"wp-image-44833\" srcset=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2-1024x256.jpg 1024w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2-300x75.jpg 300w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2-768x192.jpg 768w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2-1536x384.jpg 1536w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2-380x95.jpg 380w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2-800x200.jpg 800w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2-1160x290.jpg 1160w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2020\/11\/BANNER_EV_eletr_part1-2.jpg 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure><\/p>\n","protected":false},"excerpt":{"rendered":"O princ&iacute;pio da indu&ccedil;&atilde;o finita ou simplesmente indu&ccedil;&atilde;o &eacute; um m&eacute;todo matem&aacute;tico que pode ser utilizado para demonstrar&hellip;\n","protected":false},"author":18,"featured_media":32783,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[29],"tags":[480,285],"wl_entity_type":[732],"class_list":{"0":"post-32778","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"tag-inducao-finita","9":"tag-matematica","10":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v25.9) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Princ\u00edpio da Indu\u00e7\u00e3o Finita<\/title>\n<meta name=\"description\" content=\"Se preparando para os exames de vestibular? 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