{"id":11880,"date":"2014-05-21T22:43:06","date_gmt":"2014-05-21T21:43:06","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11880"},"modified":"2022-01-13T01:31:26","modified_gmt":"2022-01-13T01:31:26","slug":"uma-sucessao-de-termo-geral-da-forma-u_n-left-n-a-right2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11880","title":{"rendered":"Uma sucess\u00e3o de termo geral da forma ${u_n} = {\\left( {n &#8211; A} \\right)^2}$"},"content":{"rendered":"<p><ul id='GTTabs_ul_11880' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11880' class='GTTabs_curr'><a  id=\"11880_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11880' ><a  id=\"11880_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_11880'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Encontre uma sucess\u00e3o $\\left( {{u_n}} \\right)$ cujo termo geral seja da forma ${u_n} = {\\left( {n &#8211; A} \\right)^2}$, com $A$ um n\u00famero real, tal que:<\/p>\n<ol>\n<li>$\\left( {{u_n}} \\right)$ seja mon\u00f3tona;<\/li>\n<li>$\\left( {{u_n}} \\right)$ n\u00e3o seja mon\u00f3tona.<\/li>\n<\/ol>\n<p>Prove a sua conjetura.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11880' onClick='GTTabs_show(1,11880)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11880'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Encontre uma sucess\u00e3o $\\left( {{u_n}} \\right)$ cujo termo geral seja da forma ${u_n} = {\\left( {n &#8211; A} \\right)^2}$, com $A$ um n\u00famero real, tal que:<\/p>\n<ol>\n<li>$\\left( {{u_n}} \\right)$ seja mon\u00f3tona;<\/li>\n<li>$\\left( {{u_n}} \\right)$ n\u00e3o seja mon\u00f3tona.<\/li>\n<\/ol>\n<p>Prove a sua conjetura.<\/p>\n<\/blockquote>\n<p>\\[{u_n} = {\\left( {n &#8211; A} \\right)^2}\\]<\/p>\n<p><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":772,\r\n\"height\":425,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/p>\n<p>A anima\u00e7\u00e3o apresentada acima\u00a0\u00e9 bastante sugestiva para encontrar a solu\u00e7\u00e3o do problema.<\/p>\n<ol>\n<li>$\\left( {{u_n}} \\right)$ \u00e9\u00a0mon\u00f3tona em sentido lato para $A \\in \\left] { &#8211; \\infty ,\\frac{3}{2}} \\right]$.<br \/>\nA t\u00edtulo de exemplo, no caso de $A =\u00a0 &#8211; 1$: ${u_n} = {\\left( {n + 1} \\right)^2}$;<br \/>\n\u00ad<\/li>\n<li>$\\left( {{u_n}} \\right)$ \u00e9 n\u00e3o mon\u00f3tona para $A \\in \\left] {\\frac{3}{2}, + \\infty } \\right[$.<br \/>\nA t\u00edtulo de exemplo, no caso de $A = 3$: ${u_n} = {\\left( {n &#8211; 3} \\right)^2}$.<\/li>\n<\/ol>\n<p>\u00ad<\/p>\n<p><strong>PROVA<\/strong>:<\/p>\n<p>Ora,<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{{u_{n + 1}} &#8211; {u_n}}&amp; = &amp;{{{\\left( {\\left( {n &#8211; A} \\right) + 1} \\right)}^2} &#8211; {{\\left( {n &#8211; A} \\right)}^2}} \\\\<br \/>\n{}&amp; = &amp;{\\left( {{{\\left( {n &#8211; A} \\right)}^2} + 2\\left( {n &#8211; A} \\right) + 1} \\right) &#8211; {{\\left( {n &#8211; A} \\right)}^2}} \\\\<br \/>\n{}&amp; = &amp;{2n &#8211; 2A + 1}<br \/>\n\\end{array}\\]<\/p>\n<p>\u00ad<\/p>\n<p>A sucess\u00e3o $\\left( {{u_n}} \\right)$ ser\u00e1 mon\u00f3tona decrescente em sentido lato se e s\u00f3 se $2n &#8211; 2A + 1 \\leqslant 0,\\forall n \\in \\mathbb{N}$.<\/p>\n<p>Ora, a condi\u00e7\u00e3o<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{2n &#8211; 2A + 1 \\leqslant 0}&amp; \\Leftrightarrow &amp;{2n \\leqslant 2A &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{n \\leqslant A &#8211; \\frac{1}{2}}<br \/>\n\\end{array}\\]<\/p>\n<p>n\u00e3o \u00e9 universal em $\\mathbb{N}$, com $A \\in \\mathbb{R}$.<\/p>\n<p><span style=\"text-decoration: underline;\">Portanto, n\u00e3o existe qualquer $A \\in \\mathbb{R}$ para o qual a sucess\u00e3o $\\left( {{u_n}} \\right)$ seja mon\u00f3tona decrescente<\/span>.<\/p>\n<p>\u00ad\u00ad<\/p>\n<p>A sucess\u00e3o $\\left( {{u_n}} \\right)$ ser\u00e1 mon\u00f3tona crescente em sentido lato se e s\u00f3 se $2n &#8211; 2A + 1 \\geqslant 0,\\forall n \\in \\mathbb{N}$.<\/p>\n<p>Ora, a condi\u00e7\u00e3o<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{2n &#8211; 2A + 1 \\geqslant 0}&amp; \\Leftrightarrow &amp;{2n \\geqslant 2A &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{n \\geqslant A &#8211; \\frac{1}{2}}<br \/>\n\\end{array}\\]<\/p>\n<p>\u00e9 universal em $\\mathbb{N}$ para $A \\in \\left] { &#8211; \\infty ,\\frac{3}{2}} \\right]$.<\/p>\n<p>\u00ad<\/p>\n<p><span style=\"text-decoration: underline;\">Portanto, a sucess\u00e3o $\\left( {{u_n}} \\right)$ \u00e9 mon\u00f3tona crescente para $A \\in \\left] { &#8211; \\infty ,\\frac{3}{2}} \\right]$<\/span>.<\/p>\n<p><span style=\"text-decoration: underline;\">No caso de $A \\in \\left] {\\frac{3}{2}, + \\infty } \\right[$, a sucess\u00e3o $\\left( {{u_n}} \\right)$ \u00e9 n\u00e3o mon\u00f3tona<\/span>.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11880' onClick='GTTabs_show(0,11880)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Encontre uma sucess\u00e3o $\\left( {{u_n}} \\right)$ cujo termo geral seja da forma ${u_n} = {\\left( {n &#8211; A} \\right)^2}$, com $A$ um n\u00famero real, tal que: $\\left( {{u_n}} \\right)$ seja mon\u00f3tona;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19428,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,372],"tags":[422,294,431],"series":[],"class_list":["post-11880","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-sucessoes-reais","tag-11-o-ano","tag-monotonia","tag-sucessoes-reais"],"views":3160,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Duas_sucessoes.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11880","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11880"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11880\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19428"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11880"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11880"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11880"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11880"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}