{"id":11883,"date":"2014-05-22T14:38:14","date_gmt":"2014-05-22T13:38:14","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11883"},"modified":"2022-01-13T01:38:52","modified_gmt":"2022-01-13T01:38:52","slug":"prove-que-a-sucessao-e-minorada-e-nao-e-limitada","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11883","title":{"rendered":"Prove que a sucess\u00e3o \u00e9 minorada e n\u00e3o \u00e9 limitada"},"content":{"rendered":"<p><ul id='GTTabs_ul_11883' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11883' class='GTTabs_curr'><a  id=\"11883_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11883' ><a  id=\"11883_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_11883'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Em rela\u00e7\u00e3o \u00e0 sucess\u00e3o de termo geral ${a_n} = 3n + 5$, prove que \u00e9 minorada e n\u00e3o \u00e9 limitada.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11883' onClick='GTTabs_show(1,11883)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11883'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Em rela\u00e7\u00e3o \u00e0 sucess\u00e3o de termo geral ${a_n} = 3n + 5$, prove que \u00e9 minorada e n\u00e3o \u00e9 limitada.<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<p>Como ${a_{n + 1}} &#8211; {a_n} &gt; 0,\\forall n \\in \\mathbb{N}$, ent\u00e3o a sucess\u00e3o \u00e9 estritamente crescente.<\/p>\n<p>Assim, o seu primeiro termo, ${a_1} = 8$,\u00a0\u00e9 o m\u00ednimo do conjunto dos seus termos.<\/p>\n<p>Logo, a sucess\u00e3o \u00e9 minorada, pois $8 \\leqslant {a_n},\\forall n \\in \\mathbb{N}$.<\/p>\n<\/p>\n<p>Seja $M \\in {\\mathbb{R}^ + }$.<\/p>\n<p>Ora, a condi\u00e7\u00e3o<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{{a_n} &gt; M}&amp; \\Leftrightarrow &amp;{3n + 5 &gt; M} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{n &gt; \\frac{{M &#8211; 5}}{3}}<br \/>\n\\end{array}\\]<\/p>\n<p>\u00e9 poss\u00edvel para todo o $M \\in {\\mathbb{R}^ + }$.<\/p>\n<p>Isto significa que, para todo $M \\in {\\mathbb{R}^ + }$, existem termos da sucess\u00e3o que s\u00e3o superiores a $M$.<\/p>\n<p>Como $M$ pode ser qualquer n\u00famero real positivo, conclui-se que a sucess\u00e3o n\u00e3o \u00e9 majorada e, consequentemente, n\u00e3o \u00e9 limitada (ainda que seja limitada inferiormente).<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11883' onClick='GTTabs_show(0,11883)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Em rela\u00e7\u00e3o \u00e0 sucess\u00e3o de termo geral ${a_n} = 3n + 5$, prove que \u00e9 minorada e n\u00e3o \u00e9 limitada. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19189,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,372],"tags":[422,373,431],"series":[],"class_list":["post-11883","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-sucessoes-reais","tag-11-o-ano","tag-sucessao-limitada","tag-sucessoes-reais"],"views":5190,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat75.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11883","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=11883"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11883\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19189"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11883"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11883"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11883"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11883"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}