{"id":115182,"date":"2018-11-27T20:16:47","date_gmt":"2018-11-27T22:16:47","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=115182"},"modified":"2024-06-07T19:39:16","modified_gmt":"2024-06-07T22:39:16","slug":"problema-para-ajudar-na-escola-valor-da-expressao","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-valor-da-expressao\/","title":{"rendered":".Problema para ajudar na escola: Valor da express\u00e3o"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=219999\" rel=\"noopener\" target=\"_blank\">&#9999; <span style=\"color:#915F6D;font-size:14px;\">Link do problema para dispositivos da <em>Apple<\/em>.<\/span><\/a><\/strong><\/p>\n<div class=\"g1n\" style=\"line-height:25px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Problema<br \/>\n  <span style=\"font-size:12px;color:#00BF00\"> (A partir do 9\u00ba ano do E. F. &#8211; <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: M\u00e9dio<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n(<strong>ONEM 2008<\/strong> &#8211; Adaptado) Calcule o valor da express\u00e3o num\u00e9rica <\/p>\n<p style=\"margin:20px;text-align:center;\"> [tex]\\left(1^2-2^2-3^2+4^2 \\right)+ \\left(5^2-6^2-7^2+8^2 \\right)+ \\cdots + \\left(17^2-18^2-19^2+20^2 \\right)\\, .[\/tex]\n<\/p>\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:28px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nVeja que:<\/p>\n<ul>\n<li>[tex]1^2-2^2-3^2+4^2=1-4-9+16=17-13=\\boxed{4}.[\/tex]<\/li>\n<li>[tex]5^2-6^2-7^2+8^2=25-36-49+64=89-85=\\boxed{4}.[\/tex]<\/li>\n<li>[tex]9^2-10^2-11^2+12^2=81-100-121+144=225-221=\\boxed{4}.[\/tex]<\/li>\n<li>[tex]13^2-14^2-15^2+16^2=169-196-225+256=425-421=\\boxed{4}.[\/tex]<\/li>\n<li>[tex]17^2-18^2-19^2+20^2=289-324-361+400=689-685=\\boxed{4}.[\/tex]<\/li>\n<\/ul>\n<p>Assim,<br \/>\n<span style=\"color:#800000;padding-left:30px;\">&#9658;<\/span>[tex]\\left(1^2-2^2-3^2+4^2 \\right)+ \\left(5^2-6^2-7^2+8^2 \\right)+ \\cdots \\\\<br \/>\n\\qquad \\cdots + \\left(17^2-18^2-19^2+20^2 \\right)= =5 \\times \\boxed{4}=20\\, .[\/tex]<\/p>\n<p style=\"margin:40px\">\n<p class=\"ti1 c3\">Coincid\u00eancia?<\/p>\n<p class=\"ti2\">&#8211; N\u00c3O! Observe&#8230;<\/p>\n<p>Perceba que cada par\u00eantese da express\u00e3o num\u00e9rica em quest\u00e3o tem quatro parcelas, sendo a \u00faltima da forma [tex]\\left(4n\\right)^2[\/tex], com [tex]n[\/tex] um n\u00famero natural n\u00e3o nulo. Portanto, o n-\u00e9simo par\u00eantese da express\u00e3o num\u00e9rica pode ser assim definido:<br \/>\n[tex]\\qquad \\left(4n-3\\right)^2-\\left(4n-2\\right)^2-\\left(4n-1\\right)^2+\\left(4n\\right)^2.[\/tex]<br \/>\nFazendo as continhas, segue que<br \/>\n[tex]\\qquad \\qquad \\left(4n-3\\right)^2-\\left(4n-2\\right)^2-\\left(4n-1\\right)^2+\\left(4n\\right)^2 \\\\<br \/>\n\\qquad \\qquad= \\left(16n^2-24n+9\\right)-\\left(16n^2-16n+4\\right)-\\left(16n^2-8n+1\\right)+\\left(16n^2\\right)\\\\<br \/>\n\\qquad \\qquad= \\left(16n^2-16n^2-16n^2+16n^2\\right)+\\left(-24n+16n+8n\\right)+\\left(9-4-1\\right)\\\\<br \/>\n\\qquad \\qquad=0+0+4=4,[\/tex]<br \/>\nou seja, <\/p>\n<ul>\n<li>[tex]\\boxed{\\left(\\left(4n-3\\right)^2-\\left(4n-2\\right)^2-\\left(4n-1\\right)^2+\\left(4n\\right)^2\\right) =4}\\, [\/tex], para qualquer n\u00famero natural n\u00e3o nulo [tex]n[\/tex].<\/li>\n<\/ul>\n<p>Assim, para determinarmos as somas finais abaixo, por exemplo, s\u00f3 precisamos determinar quantos par\u00eanteses do tipo [tex]\\left(\\left(4n-3\\right)^2-\\left(4n-2\\right)^2-\\left(4n-1\\right)^2+\\left(4n\\right)^2\\right)\\, [\/tex] aparecem em cada soma <span style=\"font-size:12px; color:#0000FF\">(ou at\u00e9 menos do que isso)<\/span>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob8.png\" alt=\"\" width=\"800\" height=\"433\" class=\"aligncenter size-full wp-image-171518\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob8.png 860w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob8-300x163.png 300w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/fig0prob8-768x416.png 768w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2014\/09\/carinha_legal.jpg\" alt=\"\" width=\"122\" height=\"100\" class=\"aligncenter size-full wp-image-22758\" \/><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g3\">\nParticipou da discuss\u00e3o o Clube <strong>OCTETO MATEM\u00c1TICO<\/strong>.<\/div>\n<p style=\"margin:20px\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto;width:640px;\">\n<tr>\n<td class=\"bf\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1001 alignright\" style=\"border: 0px;\" alt=\"\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" width=\"80\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:520px;background-color: #FFF5EE; padding:10px; font-family: calibri; font-size: 13px;color: #1C1C1C; line-height: 20px; \"> Se for conveniente, voc\u00ea pode obter um arquivo PDF desta p\u00e1gina, com o problema e a solu\u00e7\u00e3o, clicando no bot\u00e3o abaixo.<br \/>\nVoc\u00ea pode abrir o arquivo diretamente no seu navegador (<em>Chrome<\/em>, <em>Edge<\/em>, <em>Firefox<\/em>, <em>Safari<\/em>, entre outros), mas tamb\u00e9m pode utilizar o software gratuito <em><strong>Adobe Acrobat Reader<\/strong><\/em>.<br \/>\nCaso o dispositivo que voc\u00ea est\u00e1 utilizando n\u00e3o tenha o <strong><em>Acrobat Reader<\/em><\/strong> instalado, \u00e9 s\u00f3 clicar <a href=\"http:\/\/get.adobe.com\/br\/reader\/\" rel=\"noopener\" target=\"_blank\"><strong>AQUI<\/strong><\/a> para fazer o download adequado ao seu dispositivo.<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<div style=\" float:right;  width: 100px;\">\n<p style=\"text-align: center; border-style: dotted; border-color :#FFFAFA; border-width :thin; width:90px;  height:40px;  padding: 5px; background-color: #800000;\">\n<a href=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/10\/Valor-da-express\u00e3o_.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ Valor da express\u00e3o.pdf\"><span style=\"color: #FFFAFA; font-family: georgia; font-size: 16px;\"><strong>Arquivo <\/strong><\/span><\/a>\n <\/p>\n<\/div>\n<div style=\"clear: both;\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>&#9999; Link do problema para dispositivos da Apple. Problema (A partir do 9\u00ba ano do E. F. &#8211; N\u00edvel de dificuldade: M\u00e9dio) (ONEM 2008 &#8211; Adaptado) Calcule o valor da express\u00e3o num\u00e9rica [tex]\\left(1^2-2^2-3^2+4^2 \\right)+ \\left(5^2-6^2-7^2+8^2 \\right)+ \\cdots + \\left(17^2-18^2-19^2+20^2 \\right)\\, .[\/tex] Solu\u00e7\u00e3o Veja que: [tex]1^2-2^2-3^2+4^2=1-4-9+16=17-13=\\boxed{4}.[\/tex] [tex]5^2-6^2-7^2+8^2=25-36-49+64=89-85=\\boxed{4}.[\/tex] [tex]9^2-10^2-11^2+12^2=81-100-121+144=225-221=\\boxed{4}.[\/tex] [tex]13^2-14^2-15^2+16^2=169-196-225+256=425-421=\\boxed{4}.[\/tex] [tex]17^2-18^2-19^2+20^2=289-324-361+400=689-685=\\boxed{4}.[\/tex] Assim, &#9658;[tex]\\left(1^2-2^2-3^2+4^2 \\right)+ \\left(5^2-6^2-7^2+8^2 \\right)+ \\cdots &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-valor-da-expressao\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-115182","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/115182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=115182"}],"version-history":[{"count":29,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/115182\/revisions"}],"predecessor-version":[{"id":220001,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/115182\/revisions\/220001"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=115182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}