{"id":11890,"date":"2014-05-22T17:25:55","date_gmt":"2014-05-22T16:25:55","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11890"},"modified":"2022-01-13T10:45:58","modified_gmt":"2022-01-13T10:45:58","slug":"estude-a-monotonia-da-sucessao-definida-por-recorrencia","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11890","title":{"rendered":"Estude a monotonia da sucess\u00e3o definida por recorr\u00eancia"},"content":{"rendered":"<p><ul id='GTTabs_ul_11890' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11890' class='GTTabs_curr'><a  id=\"11890_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11890' ><a  id=\"11890_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_11890'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Estude a monotonia da sucess\u00e3o definida por recorr\u00eancia:<\/p>\n<p>\\[\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{{r_1} = 1} \\\\<br \/>\n{{r_n} = \\frac{{{r_{n &#8211; 1}}}}{2},n \\geqslant 2}<br \/>\n\\end{array}} \\right.\\]<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11890' onClick='GTTabs_show(1,11890)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11890'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Estude a monotonia da sucess\u00e3o definida por recorr\u00eancia:<\/p>\n<p>\\[\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{{r_1} = 1} \\\\<br \/>\n{{r_n} = \\frac{{{r_{n &#8211; 1}}}}{2},n \\geqslant 2}<br \/>\n\\end{array}} \\right.\\]<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<p>A sucess\u00e3o pode, tamb\u00e9m, ser definida por:<\/p>\n<p>\\[\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{{r_1} = 1} \\\\<br \/>\n{{r_{n + 1}} = \\frac{{{r_n}}}{2},\\forall n \\in \\mathbb{N}}<br \/>\n\\end{array}} \\right.\\]<\/p>\n<p>Antes de mais, conv\u00e9m reparar que todo o termo da sucess\u00e3o \u00e9 um n\u00famero positivo, pois o primeiro termo \u00e9 $1$ e, a partir deste, o termo consecutivo \u00e9 metade do anterior (o que permite concluir que a sucess\u00e3o \u00e9 estritamente decrescente).<\/p>\n<p>Para todo o ${n \\in \\mathbb{N}}$ e de acordo com o estabelecido na defini\u00e7\u00e3o por recorr\u00eancia de $\\left( {{r_n}} \\right)$, vem:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{{r_{n + 1}} = \\frac{{{r_n}}}{2}}&amp; \\Leftrightarrow &amp;{{r_{n + 1}} &#8211; \\frac{{{r_n}}}{2} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{2 \\times {r_{n + 1}} &#8211; {r_n} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{r_{n + 1}} + {r_{n + 1}} &#8211; {r_n} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\boxed{{r_{n + 1}} &#8211; {r_n} =\u00a0 &#8211; {r_{n + 1}}}}<br \/>\n\\end{array}\\]<\/p>\n<p>Como todo termo da sucess\u00e3o \u00e9 positivo, ent\u00e3o conclui-se que ${r_{n + 1}} &#8211; {r_n} &lt; 0,\\forall n \\in \\mathbb{N}$, isto \u00e9, que sucess\u00e3o \u00e9 estritamente decrescente.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11890' onClick='GTTabs_show(0,11890)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Estude a monotonia da sucess\u00e3o definida por recorr\u00eancia: \\[\\left\\{ {\\begin{array}{*{20}{l}} {{r_1} = 1} \\\\ {{r_n} = \\frac{{{r_{n &#8211; 1}}}}{2},n \\geqslant 2} \\end{array}} \\right.\\] Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19178,"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-11890","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":10852,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat69.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11890","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=11890"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11890\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19178"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11890"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11890"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11890"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11890"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}