{"id":20229,"date":"2014-06-20T02:30:31","date_gmt":"2014-06-20T05:30:31","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=20229"},"modified":"2023-02-19T14:39:35","modified_gmt":"2023-02-19T17:39:35","slug":"problemao-raiz-de-raizes","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-raiz-de-raizes\/","title":{"rendered":".Problem\u00e3o: Raiz de ra\u00edzes"},"content":{"rendered":"

\n

\n

Problema
\n (Indicado a partir do 8\u00ba ano do E. F.)<\/span><\/p>\n

\n

\n

Determine o valor da express\u00e3o num\u00e9rica
\n[tex]\\qquad \\qquad \\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{1+\\cdots}}}}\\,.[\/tex]<\/p><\/div>\n

\n

\n

Solu\u00e7\u00e3o<\/p>\n

\n

\n

Observe que
\n[tex]\\qquad 3^2=1+8=1+2\\times 4[\/tex];
\n[tex]\\qquad 4^2=1+15=1+3\\times 5[\/tex];
\n[tex]\\qquad 5^2=1+24=1+4\\times 6[\/tex];
\n[tex]\\qquad 6^2=1+35=1+5\\times 7[\/tex].
\nAssim,
\n[tex]\\qquad 3^2=1+2\\times \\sqrt{1+3\\times 5}=1+2\\times \\sqrt{1+3\\times \\sqrt{1+4\\times 6}}\\\\
\n\\qquad 3^2=1+2\\times \\sqrt{1+3\\times \\sqrt{1+4\\times \\sqrt{1+\\cdots}}}\\;\\;[\/tex]
\ne, consequentemente,
\n[tex]\\qquad 3=\\sqrt{1+2\\times \\sqrt{1+3\\times \\sqrt{1+4\\times \\sqrt{1+\\cdots}}}}\\;\\; .[\/tex]
\nDessa forma, o resultado final da express\u00e3o \u00e9 [tex] \\,\\fcolorbox{black}{#eee0e5}{$3$}\\,[\/tex].<\/p>\n

\n

Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong> .<\/p>\n<\/div>\n

\n

Participou da discuss\u00e3o o Clube Matem\u00e1tica em constru\u00e7\u00e3o<\/strong>. <\/div>\n","protected":false},"excerpt":{"rendered":"

Problema (Indicado a partir do 8\u00ba ano do E. F.) Determine o valor da express\u00e3o num\u00e9rica [tex]\\qquad \\qquad \\sqrt{1+2\\sqrt{1+3\\sqrt{1+4\\sqrt{1+\\cdots}}}}\\,.[\/tex] Solu\u00e7\u00e3o Observe que [tex]\\qquad 3^2=1+8=1+2\\times 4[\/tex]; [tex]\\qquad 4^2=1+15=1+3\\times 5[\/tex]; [tex]\\qquad 5^2=1+24=1+4\\times 6[\/tex]; [tex]\\qquad 6^2=1+35=1+5\\times 7[\/tex]. Assim, [tex]\\qquad 3^2=1+2\\times \\sqrt{1+3\\times 5}=1+2\\times \\sqrt{1+3\\times \\sqrt{1+4\\times 6}}\\\\ \\qquad 3^2=1+2\\times \\sqrt{1+3\\times \\sqrt{1+4\\times \\sqrt{1+\\cdots}}}\\;\\;[\/tex] e, consequentemente, [tex]\\qquad 3=\\sqrt{1+2\\times \\sqrt{1+3\\times \\sqrt{1+4\\times \\sqrt{1+\\cdots}}}}\\;\\; .[\/tex] … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-20229","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/20229","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\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=20229"}],"version-history":[{"count":26,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/20229\/revisions"}],"predecessor-version":[{"id":183943,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/20229\/revisions\/183943"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=20229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}