{"id":33181,"date":"2015-03-19T21:29:12","date_gmt":"2015-03-20T00:29:12","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=33181"},"modified":"2023-03-21T21:22:51","modified_gmt":"2023-03-22T00:22:51","slug":"problema-sucessao-de-raizes","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-sucessao-de-raizes\/","title":{"rendered":".Problema: Sucess\u00e3o de ra\u00edzes"},"content":{"rendered":"

\n

\n

Problema
\n (Indicado a partir do 9\u00ba ano do E. F.)<\/span><\/p>\n

\n

\n

Qual o valor de [tex]\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+ \\sqrt{2+…}}}}}}[\/tex]  ?\n<\/p><\/div>\n

\n

\n

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

\nSeja [tex]\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+…}}}}}=k[\/tex].
\nEnt\u00e3o, observe que <\/p>\n

[tex]\\quad k=\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+…}}}}}=\\sqrt{2+k}[\/tex],
\ndonde
\n[tex]\\quad k^2=2+k[\/tex].
\nAssim, obtemos a equa\u00e7\u00e3o do segundo grau [tex] k^2-k-2=0 [\/tex], cujas ra\u00edzes s\u00e3o [tex]k_1=-1\\,[\/tex] e [tex]\\,k_2=2[\/tex].<\/p>\n

Como nossa express\u00e3o inicial define apenas valores positivos, temos que <\/p>\n

[tex]\\quad k=\\,\\fcolorbox{black}{#eee0e5}{$\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+…}}}}}=2$}\\,.[\/tex]<\/p>\n

\n

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

\n","protected":false},"excerpt":{"rendered":"

Problema (Indicado a partir do 9\u00ba ano do E. F.) Qual o valor de [tex]\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+ \\sqrt{2+…}}}}}}[\/tex]  ? Solu\u00e7\u00e3o Seja [tex]\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+…}}}}}=k[\/tex]. Ent\u00e3o, observe que [tex]\\quad k=\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+…}}}}}=\\sqrt{2+k}[\/tex], donde [tex]\\quad k^2=2+k[\/tex]. Assim, obtemos a equa\u00e7\u00e3o do segundo grau [tex] k^2-k-2=0 [\/tex], cujas ra\u00edzes s\u00e3o [tex]k_1=-1\\,[\/tex] e [tex]\\,k_2=2[\/tex]. Como nossa express\u00e3o inicial define apenas valores positivos, temos que [tex]\\quad … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":165,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-33181","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/33181","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\/165"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=33181"}],"version-history":[{"count":17,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/33181\/revisions"}],"predecessor-version":[{"id":185036,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/33181\/revisions\/185036"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=33181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}