Problema Para quais valores de [tex]k[\/tex] a sequ\u00eancia [tex]S_k[\/tex] cont\u00e9m o termo 2005?<\/p>\n \n<\/div>\n \n Solu\u00e7\u00e3o <\/p>\n Note que isso quer dizer que \n[tex]\\begin{array}{|l|l|c|} Assim, os poss\u00edveis valores para [tex]k[\/tex] s\u00e3o: Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n \n Tentando entender…<\/p>\n \n \n[tex]\\begin{array}{|l|l|c|} \n Melhorou?<\/p>\n \n","protected":false},"excerpt":{"rendered":" Problema (Indicado a partir do 2\u00ba ano do E. M.) Para cada inteiro positivo [tex]k[\/tex], denote por [tex]S_k[\/tex] a progress\u00e3o aritm\u00e9tica crescente de inteiros cujo primeiro termo \u00e9 1 e cuja diferen\u00e7a entre dois termos consecutivos \u00e9 [tex]k[\/tex]. Por exemplo, [tex]S_3[\/tex] \u00e9 a sequ\u00eancia [tex]1,4,7,10,\\ldots \\, [\/tex]; [tex]S_5[\/tex] \u00e9 a sequ\u00eancia [tex]1,6,11,16,\\ldots \\, \\, [\/tex]. … <\/p>\n
\n (Indicado a partir do 2\u00ba ano do E. M.)<\/span><\/p>\n
\nPara cada inteiro positivo [tex]k[\/tex], denote por [tex]S_k[\/tex] a progress\u00e3o aritm\u00e9tica crescente de inteiros cujo primeiro termo \u00e9 1 e cuja diferen\u00e7a entre dois termos consecutivos \u00e9 [tex]k[\/tex].
\nPor exemplo,<\/p>\n\n
\nSeja [tex]S_k[\/tex] uma das progress\u00f5es que satisfazem as exig\u00eancias do problema e suponha que [tex]2005[\/tex] seja o n-\u00e9simo termo dessa sequ\u00eancia.<\/p>\n![\"f022\"](\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2016\/12\/f022.png\")
<\/p>\n
\n[tex]\\qquad \\qquad 1+(n-1)k=2005[\/tex]
\ne, portanto, temos que:
\n[tex]\\qquad \\qquad k(n-1)=2004[\/tex].
\nMas [tex]2004=2^2\\times 3\\times 167[\/tex], ent\u00e3o
\n[tex]\\qquad \\qquad k(n-1)=2^2\\times3\\times167. \\qquad (i)[\/tex]
\nNa tabela abaixo, podemos ver os inteiros positivos [tex]k \\, [\/tex] e [tex] \\, n-1[\/tex] que satisfazem a equa\u00e7\u00e3o [tex] (i) \\, [\/tex].<\/p>\n
\n\\hline k & n-1& k\\times(n-1) \\\\
\n\\hline 1& 2004&2004\\\\
\n\\hline 2& 1002&2004\\\\
\n\\hline 3& 668&2004\\\\
\n\\hline 4& 501&2004\\\\
\n\\hline 6& 334&2004\\\\
\n\\hline 12& 167&2004\\\\
\n\\hline 167& 12&2004\\\\
\n\\hline 334& 6&2004\\\\
\n\\hline 501& 4&2004\\\\
\n\\hline 668& 3&2004\\\\
\n\\hline 1002& 2&2004\\\\
\n\\hline 2004& 1&2004\\\\
\n \\hline
\n\\end{array}[\/tex]<\/p>\n
\n[tex]\\qquad 1, \\, \\, 2, \\, \\, 3, \\, \\, 4, \\, \\, 6, \\, \\, 12, \\, \\, 167, \\, \\, 334, \\, \\, 501, \\, \\, 668, \\, \\, 1002 \\, \\, [\/tex] e [tex] \\, \\, 2004[\/tex].<\/p>\n
\n
\nSe voc\u00ea n\u00e3o entendeu de onde apareceram os valores de [tex]k \\, [\/tex] e [tex] \\, n-1[\/tex], a tabela abaixo pode ajudar.<\/p>\n
\n\\hline k & n-1& k\\times(n-1) \\\\
\n\\hline \\boxed{1}& 2^2\\times 3\\times 167=\\boxed{2004} &2004\\\\
\n\\hline \\boxed{2}& 2\\times 3\\times 167= \\boxed{1002}&2004\\\\
\n\\hline \\boxed{3}& 2^2\\times 167=\\boxed{668} &2004\\\\
\n\\hline 2^2=\\boxed{4}& 3 \\times 167=\\boxed{501} &2004\\\\
\n\\hline 2 \\times 3=\\boxed{6} & 2\\times 167=\\boxed{334}&2004\\\\
\n\\hline 2^2\\times 3=\\boxed{12}& \\boxed{167}&2004\\\\
\n\\hline \\boxed{167}& 2^2 \\times 3=\\boxed{12} &2004\\\\
\n\\hline 2\\times 167=\\boxed{334} & 2 \\times 3=\\boxed{6}&2004\\\\
\n\\hline 3 \\times 167=\\boxed{501}& 2^2=\\boxed{4}&2004\\\\
\n\\hline 2^2\\times 167=\\boxed{668}& \\boxed{3}&2004\\\\
\n\\hline 2\\times 3\\times 167=\\boxed{1002}& \\boxed{2}&2004\\\\
\n\\hline 2^2\\times 3\\times 167=\\boxed{2004} & \\boxed{1}&2004\\\\
\n \\hline
\n\\end{array}[\/tex]<\/p>\n![\"carinha1\"](\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/02\/carinha1.jpg\")
\n<\/div>\n