{"id":147705,"date":"2020-03-02T15:52:49","date_gmt":"2020-03-02T18:52:49","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=147705"},"modified":"2023-06-10T23:40:06","modified_gmt":"2023-06-11T02:40:06","slug":"problemao-soma-de-inteiros-consecutivos","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-soma-de-inteiros-consecutivos\/","title":{"rendered":".Problem\u00e3o: Soma de inteiros consecutivos"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:25px;\">\n<p class=\"tit\"style=\"line-height:20px; margin-bottom: 10px;\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 1\u00ba ano do E. M.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p><span style=\"font-size:12px\">(<strong>The Ussr Olympiad Problem Book<\/strong> &#8211; Adaptado)<\/span> Encontre sequ\u00eancias de inteiros positivos consecutivos cuja soma seja igual a [tex]1000[\/tex].<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<div class=\"cb36 g1\" style=\"margin:40px;line-height:38px; width:60%; margin-left:auto; margin-right:auto; \"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" style=\"border:none\"  alt=\"explicador_p\" width=\"70\" height=\"84\" class=\"alignleft size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 ti3\"style=\"line-height:20px; margin-bottom: 10px;\"> Lembrete<\/p>\n<p style=\"margin:10px\">\n<p> Se [tex]a_1, a_2, \\cdots, a_n[\/tex] for uma progress\u00e3o aritm\u00e9tica de raz\u00e3o [tex]r[\/tex], ent\u00e3o a soma dos seus [tex]n[\/tex] termos \u00e9 dada por [tex]\\,\\boxed{S_n=\\dfrac{(a_1+a_n)\\cdot n}{2}}\\,.[\/tex]<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<div class=\"g1n\" style=\"line-height:28px;\">\n<p class=\"tit\"style=\"line-height:30px; margin-bottom: 10px;\">Solu\u00e7\u00e3o<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>Suponha que [tex]\\,\\, m,\\, m+1, \\,m+2, \\cdots, \\,m+k\\,\\,[\/tex] sejam inteiros positivos e consecutivos cuja soma \u00e9 [tex]1000[\/tex].<br \/>\nObserve que esses n\u00fameros formam uma progress\u00e3o aritm\u00e9tica de raz\u00e3o [tex]1[\/tex]. Assim, como a soma desses [tex]k+1[\/tex] n\u00fameros \u00e9 [tex]1000[\/tex], pelo <strong>Lembrete <\/strong> segue que:<\/p>\n<p style=\"margin:10px\">\n[tex]\\quad \\dfrac{(m+(m+k))\\cdot (k+1)}{2}=1000[\/tex]<br \/>\n[tex]\\quad (2m+k)\\cdot(k+1)=2000.\\ \\ \\ \\ \\ \\ \\ \\ \\textcolor{#800000}{(i)}[\/tex]<\/p>\n<p>Sendo [tex]2m[\/tex] um n\u00famero par, temos que:<\/p>\n<ul>\n<li> se [tex]k[\/tex] for <span class=\"c3\"><strong>\u00edmpar<\/strong><\/span>, ent\u00e3o [tex]2m+k[\/tex] \u00e9 \u00edmpar e [tex]k+1[\/tex] \u00e9 par;<\/li>\n<li> se [tex]k[\/tex] for par, ent\u00e3o [tex]2m+k[\/tex] \u00e9 par e [tex]k+1[\/tex] \u00e9 <span class=\"c3\"><strong>\u00edmpar<\/strong><\/span>.<\/li>\n<\/ul>\n<p>Dessa forma, em ambos os casos um dos fatores do lado esquerdo da igualdade [tex]\\textcolor{#800000}{(i)}[\/tex] \u00e9 <span class=\"c3\"><strong>\u00edmpar<\/strong><\/span>. Observe tamb\u00e9m que, como [tex]m[\/tex] \u00e9 um inteiro positivo, ent\u00e3o [tex]2m+k[\/tex] \u00e9 maior que [tex]k+1[\/tex].<br \/>\nPor outro lado, temos que [tex]2000=2^4\\cdot 5^3[\/tex], logo as \u00fanicas maneiras de escrever [tex]2000[\/tex] como produto de dois fatores sendo um deles <span class=\"c3\"><strong>\u00edmpar<\/strong><\/span> s\u00e3o:<br \/>\n[tex]\\quad \\textcolor{#800000}{(a)}\\,\\, 2000=2000\\cdot 1[\/tex];<br \/>\n[tex]\\quad \\textcolor{#800000}{(b)}\\,\\, 2000=400\\cdot 5[\/tex];<br \/>\n[tex]\\quad \\textcolor{#800000}{(c)}\\,\\, 2000=125\\cdot 16[\/tex];<br \/>\n[tex]\\quad \\textcolor{#800000}{(d)}\\,\\, 2000=80\\cdot 25[\/tex];<br \/>\ndonde segue, respectivamente, que:<br \/>\n[tex]\\quad \\textcolor{#800000}{(a)}\\,\\, 2m+k=2000[\/tex] e [tex]k+1=1[\/tex], ent\u00e3o [tex]k=0[\/tex] e [tex]m=1000[\/tex];<br \/>\n[tex]\\quad \\textcolor{#800000}{(b)}\\,\\, 2m+k=400[\/tex] e [tex]k+1=5[\/tex], ent\u00e3o [tex]k=4[\/tex] e [tex]m=198[\/tex];<br \/>\n[tex]\\quad \\textcolor{#800000}{(c)}\\,\\, 2m+k=125[\/tex] e [tex]k+1=16[\/tex], ent\u00e3o [tex]k=15[\/tex] e [tex]m=55[\/tex];<br \/>\n[tex]\\quad \\textcolor{#800000}{(d)}\\,\\, 2m+k=80[\/tex] e [tex]k+1=25[\/tex], ent\u00e3o [tex]k=24[\/tex] e [tex]m=28[\/tex].<br \/>\nPortanto, as sequ\u00eancias de inteiros positivos consecutivos cuja soma \u00e9 igual a [tex]1000[\/tex] s\u00e3o:<br \/>\n[tex]\\quad \\textcolor{#800000}{(a)}\\,\\,[\/tex] Sequ\u00eancia com um elemento:<br \/>\n[tex]\\quad \\qquad \\quad \\textcolor{red}{\\ggg} 1000[\/tex].<span style=\"font-size:12px;\">(Sequ\u00eancias com apenas um elemento s\u00e3o matematicamente consideradas)<\/span><br \/>\n[tex]\\quad \\textcolor{#800000}{(b)}\\,\\,[\/tex] Sequ\u00eancia com cinco elementos:<br \/>\n[tex]\\quad \\qquad \\quad \\textcolor{red}{\\ggg} 198, 199, 200, 201[\/tex] e [tex]202[\/tex].<br \/>\n[tex]\\quad \\textcolor{#800000}{(c)}\\,\\,[\/tex] Sequ\u00eancia com dezesseis elementos:<br \/>\n[tex]\\quad \\qquad \\quad \\textcolor{red}{\\ggg} 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69[\/tex] e [tex]70[\/tex].<br \/>\n[tex]\\quad \\textcolor{#800000}{(d)}\\,\\,[\/tex] Sequ\u00eancia com vinte e cinco elementos:<br \/>\n[tex]\\quad \\qquad \\quad \\textcolor{red}{\\ggg} 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,\\\\<br \/>\n \\quad \\qquad \\quad \\quad 46, 47, 48, 49, 50, 51 \\text{ e }52[\/tex]. <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:10px\">\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 1\u00ba ano do E. M.) (The Ussr Olympiad Problem Book &#8211; Adaptado) Encontre sequ\u00eancias de inteiros positivos consecutivos cuja soma seja igual a [tex]1000[\/tex]. Lembrete Se [tex]a_1, a_2, \\cdots, a_n[\/tex] for uma progress\u00e3o aritm\u00e9tica de raz\u00e3o [tex]r[\/tex], ent\u00e3o a soma dos seus [tex]n[\/tex] termos \u00e9 dada por [tex]\\,\\boxed{S_n=\\dfrac{(a_1+a_n)\\cdot n}{2}}\\,.[\/tex] Solu\u00e7\u00e3o &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-soma-de-inteiros-consecutivos\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":721,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-147705","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/147705","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\/721"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=147705"}],"version-history":[{"count":10,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/147705\/revisions"}],"predecessor-version":[{"id":194266,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/147705\/revisions\/194266"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=147705"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}