{"id":527758,"date":"2020-07-10T10:00:00","date_gmt":"2020-07-10T13:00:00","guid":{"rendered":"https:\/\/www.estrategiaconcursos.com.br\/blog\/?p=527758"},"modified":"2022-10-06T15:03:37","modified_gmt":"2022-10-06T18:03:37","slug":"progressao-aritmetica-e-progressao-geometrica-aplicacao-em-concursos","status":"publish","type":"post","link":"https:\/\/www.estrategiaconcursos.com.br\/blog\/progressao-aritmetica-e-progressao-geometrica-aplicacao-em-concursos\/","title":{"rendered":"Progress\u00e3o Aritm\u00e9tica e Progress\u00e3o Geom\u00e9trica  \u2013 Aplica\u00e7\u00e3o em concursos"},"content":{"rendered":"\n<p>Ol\u00e1, pessoal, no artigo de hoje trataremos sobre as aplica\u00e7\u00f5es de <strong>Progress\u00e3o Aritm\u00e9tica e Progress\u00e3o Geom\u00e9trica <\/strong>nas provas de concurso. Vamos l\u00e1?<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-definicao\">Defini\u00e7\u00e3o<\/h2>\n\n\n\n<p>Progress\u00e3o nada mais \u00e9 que o desenvolvimento de uma <strong>sequ\u00eancia <\/strong>que segue uma lei de forma\u00e7\u00e3o (uma f\u00f3rmula matem\u00e1tica). No nosso cotidiano estamos diante de v\u00e1rias sequ\u00eancias, tais como:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Anivers\u00e1rios: (20 anos, 21 anos, 22 anos e etc.)<\/li><li>Anos de copa do mundo: (1994, 1998, 2002 e etc.)<\/li><li>Sequ\u00eancia dos n\u00fameros naturais: (0, 1, 2, 3, 4 e etc.)<\/li><\/ul>\n\n\n\n<p>Dentro desse contexto temos a Progress\u00e3o Aritm\u00e9tica e Progress\u00e3o Geom\u00e9trica que exploraremos a seguir.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-progressao-aritmetica-pa\">Progress\u00e3o aritm\u00e9tica (PA)<\/h3>\n\n\n\n<p>A <strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">progress\u00e3o aritm\u00e9tica<\/span><\/strong> \u00e9 uma sequ\u00eancia num\u00e9rica que corresponde, a partir do segundo termo, \u00e0 <strong>soma<\/strong> do termo anterior com uma constante. Essa constante chamamos de raz\u00e3o (<strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">r<\/span><\/strong>).<\/p>\n\n\n\n<p class=\"has-text-align-center\">PA (a; a+r; a + 2r; &#8230;)<\/p>\n\n\n\n<p>Imaginemos que temos um termo inicial <img loading=\"lazy\" decoding=\"async\" width=\"13\" height=\"17\" src=\"\"\/>&nbsp;que valha 1 e que a raz\u00e3o seja de 3, teremos a seguinte sequ\u00eancia:<\/p>\n\n\n\n<p class=\"has-text-align-center\">{1; 4; 7; &#8230;}<\/p>\n\n\n\n<p>Bem tranquilo, n\u00e3o \u00e9 mesmo? Mas e se quisermos saber o 99\u00ba termo? Seria muito trabalhoso realizar essa sequ\u00eancia \u201cna m\u00e3o\u201d, por isso a necessidade de conhecermos as f\u00f3rmulas que exploraremos \u00e0 frente.<\/p>\n\n\n\n<p>Al\u00e9m disso, a P.A poder\u00e1 ser classificada como:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Crescente:<\/strong> (1; 2; 3&#8230;) r = +1<\/li><li><strong>Decrescente<\/strong>: (1; 0; -1) -&gt; r = -1<\/li><li><strong>Constante<\/strong>: (1; 1; 1) -&gt; r = 0<\/li><\/ul>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-amber-color\"><strong>Obs.<\/strong> <\/mark>A classifica\u00e7\u00e3o nem \u00e9 t\u00e3o importante assim, mas devemos observar que os valores poder\u00e3o ser positivos ou negativos a depender do caso.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-progressao-geometrica-pg\">Progress\u00e3o geom\u00e9trica (PG)<\/h3>\n\n\n\n<p>J\u00e1 a <strong><span class=\"has-inline-color has-vivid-green-cyan-color\">progress\u00e3o geom\u00e9trica<\/span><\/strong> \u00e9 uma sequ\u00eancia num\u00e9rica que corresponde, a partir do segundo termo, \u00e0 <strong>multiplica\u00e7\u00e3o<\/strong> do termo anterior pela raz\u00e3o (<strong><span class=\"has-inline-color has-vivid-green-cyan-color\">q<\/span><\/strong>). <\/p>\n\n\n\n<p class=\"has-text-align-center\">PG [a; a.q; a.q\u00b2; &#8230;]<\/p>\n\n\n\n<p>Vamos utilizar os mesmos n\u00fameros do exemplo anterior para determinar a sequ\u00eancia utilizando a PG.<\/p>\n\n\n\n<p class=\"has-text-align-center\">{1; 3; 9; &#8230;}<\/p>\n\n\n\n<p>Referente \u00e0s classifica\u00e7\u00f5es, al\u00e9m das que vimos na PA, a PG tamb\u00e9m poder\u00e1 ser <strong>oscilante<\/strong>, quando a raz\u00e3o (q) for menor que 0.<\/p>\n\n\n\n<p class=\"has-text-align-center\">{1; -2; 4; -8; 16;&#8230;}<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-formulas-pa\">F\u00f3rmulas PA<\/h2>\n\n\n\n<p>Agora que j\u00e1 definimos o conceito de PA vejamos as principais f\u00f3rmulas associadas ao tema.<\/p>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">Defini\u00e7\u00e3o:<\/span> <\/strong>J\u00e1 definimos a PA, mas matematicamente ela pode ser descrita.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"135\" height=\"58\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17105341\/image-229.png\" alt=\"Defini\u00e7\u00e3o PA\" class=\"wp-image-527767\"\/><\/figure>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">Termo Geral:<\/span> <\/strong>Trata-se da f\u00f3rmula mais importante, essa <strong>deve ser decorada<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"179\" height=\"43\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17105406\/image-230.png\" alt=\"Termo Geral PA\" class=\"wp-image-527768\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>Raz\u00e3o<\/strong>: <\/mark>A raz\u00e3o pode ser definida pela diferen\u00e7a entre dois termos consecutivos.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"127\" height=\"42\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17105428\/image-231.png\" alt=\"Raz\u00e3o PA\" class=\"wp-image-527769\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>Soma dos termos<\/strong>:<\/mark><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"156\" height=\"58\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17105449\/image-232.png\" alt=\"Soma dos termos PA\" class=\"wp-image-527770\"\/><\/figure>\n\n\n\n<p>Essa f\u00f3rmula pode ser bem \u00fatil para encontrar a soma dos termos da sequ\u00eancia.<\/p>\n\n\n\n<p>Voltando ao nosso exemplo de PA (a1 = 1 e r = 3)<img loading=\"lazy\" decoding=\"async\" width=\"31\" height=\"17\" src=\"\"\/>, vamos encontrar a soma dos cinco primeiros termos.<\/p>\n\n\n\n<p>Primeiro determinaremos <img loading=\"lazy\" decoding=\"async\" width=\"13\" height=\"17\" src=\"\"\/>o quinto termo.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"175\" height=\"103\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17105713\/image-234.png\" alt=\"\" class=\"wp-image-527772\"\/><\/figure>\n\n\n\n<p>&nbsp;Agora basta aplicar a f\u00f3rmula<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"180\" height=\"154\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17105801\/image-235.png\" alt=\"\" class=\"wp-image-527775\"\/><\/figure>\n\n\n\n<p>Veja que \u00e9 muito mais f\u00e1cil que encontrar toda a sequ\u00eancia e som\u00e1-la termo a termo.<\/p>\n\n\n\n<p class=\"has-text-align-center\">\u2211{ 1; 4; 7; 10; 13} = 35<\/p>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">Equival\u00eancia dos termos da PA: <\/span><\/strong>Trata-se de uma propriedade interessante da PA, em que:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"147\" height=\"42\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17123305\/image-238.png\" alt=\"Equival\u00eancia PA\" class=\"wp-image-527861\"\/><\/figure>\n\n\n\n<p><em>Onde,<\/em><\/p>\n\n\n\n<p><em>n + m = p + t<\/em><\/p>\n\n\n\n<p>Utilizando os valores da sequ\u00eancia anteriormente vista {1; <span class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>4<\/strong>;<\/span><span class=\"has-inline-color has-vivid-purple-color\"> <strong>7; 10<\/strong><\/span>; <strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">13<\/span><\/strong>}, teremos que:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"159\" height=\"115\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17123500\/image-239.png\" alt=\"\" class=\"wp-image-527862\"\/><\/figure>\n\n\n\n<p>Essa f\u00f3rmula \u00e9 bem \u00fatil para determinar um termo a partir de outros termos dados pelo exerc\u00edcio, sem conhecer a raz\u00e3o, por exemplo.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-formulas-pg\">F\u00f3rmulas PG<\/h2>\n\n\n\n<p>As principais f\u00f3rmulas para a Progress\u00e3o Geom\u00e9trica s\u00e3o:<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-green-cyan-color\"><strong>Defini\u00e7\u00e3o<\/strong>:<\/mark><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"145\" height=\"48\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17174652\/image-275.png\" alt=\"defini\u00e7\u00e3o PG\" class=\"wp-image-528162\"\/><\/figure>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-green-cyan-color\">Termo Geral:<\/span><\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"145\" height=\"48\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17123746\/image-241.png\" alt=\"Termo geral PG\" class=\"wp-image-527864\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-green-cyan-color\"><strong>Raz\u00e3o<\/strong>: <\/mark>A raz\u00e3o pode ser encontrada pela <strong>divis\u00e3o <\/strong>entre dois termos consecutivos.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"111\" height=\"67\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17123851\/image-242.png\" alt=\"Raz\u00e3o PG\" class=\"wp-image-527865\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-green-cyan-color\"><strong>Soma dos termos<\/strong>:<\/mark><\/p>\n\n\n\n<p>Aqui temos uma diferen\u00e7a em rela\u00e7\u00e3o \u00e0 P.A, pois podemos encontrar a soma de \u201cn\u201d termos da mesma forma que a PA, por\u00e9m tamb\u00e9m podemos encontrar a soma infinita dos termos, desde que a raz\u00e3o esteja entre -1 e 1.<\/p>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-purple-color\">Soma finita dos termos:<\/span><\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"135\" height=\"63\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17123921\/image-243.png\" alt=\"Soma dos termos finitos PG\" class=\"wp-image-527866\"\/><\/figure>\n\n\n\n<p>Vamos encontrar a soma dos cinco primeiros termos da PG, considerando que &nbsp;a1 = 1 e r = 3.<\/p>\n\n\n\n<p>O quinto termo ser\u00e1:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"142\" height=\"99\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17123948\/image-244.png\" alt=\"\" class=\"wp-image-527867\"\/><\/figure>\n\n\n\n<p>Ent\u00e3o,<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"155\" height=\"225\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17124105\/image-246.png\" alt=\"\" class=\"wp-image-527870\"\/><\/figure>\n\n\n\n<p>Confirmando,<\/p>\n\n\n\n<p class=\"has-text-align-center\">\u2211{1; 3; 9;27; 81} = 121<\/p>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-purple-color\">Soma infinita dos termos:<\/span><\/strong><\/p>\n\n\n\n<p>Para somas de infinitos termos utilizamos a seguinte f\u00f3rmula (aten\u00e7\u00e3o \u00e0 condi\u00e7\u00e3o!):<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"116\" height=\"66\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17124337\/image-247.png\" alt=\"Soma dos termos infinitos PG\" class=\"wp-image-527871\"\/><\/figure>\n\n\n\n<p><strong>Condi\u00e7\u00e3o: <\/strong>-1 &lt; q &lt; 1<\/p>\n\n\n\n<p>Exemplo num\u00e9rico:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"70\" height=\"57\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17175147\/image-276.png\" alt=\"\" class=\"wp-image-528163\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"130\" height=\"180\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17124500\/image-248.png\" alt=\"\" class=\"wp-image-527872\"\/><\/figure>\n\n\n\n<p>E existe uma explica\u00e7\u00e3o? Claro, ao avan\u00e7armos na sequ\u00eancia (n tendendo ao infinito) em que a raz\u00e3o (q) atende \u00e0 condi\u00e7\u00e3o -1 &lt; q &lt; 1, os valores ser\u00e3o cada vez menores, <strong>tendendo a zero<\/strong>, dessa forma \u00e9 poss\u00edvel determinar o valor da soma, essa \u00e9 a ideia de <span style=\"text-decoration: underline\">limite<\/span>. <\/p>\n\n\n\n<p class=\"has-text-align-center\">{ 5; 2,5; 1,25; 0,625; 0,3125; 0,15625; 0,078125; &#8230;.}<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-green-cyan-color\"><b>Equival\u00eancia dos termos da PG: <\/b><\/mark>Assim como na PA, \u00e9 poss\u00edvel determinar um termo a partir de sua \u201cdist\u00e2ncia\/posi\u00e7\u00e3o\u201d, por\u00e9m aten\u00e7\u00e3o, pois na PG encontraremos a partir do <strong>produto<\/strong> e n\u00e3o da soma. <\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"155\" height=\"57\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17125157\/image-253.png\" alt=\"Equival\u00eancia PG\" class=\"wp-image-527879\"\/><\/figure>\n\n\n\n<p><em>Onde,<\/em><\/p>\n\n\n\n<p><em>n + m = p + t<\/em><\/p>\n\n\n\n<p>Utilizando os n\u00fameros de nossa sequ\u00eancia {1; <strong><span class=\"has-inline-color has-vivid-green-cyan-color\">3<\/span><\/strong>; <strong><span class=\"has-inline-color has-vivid-purple-color\">9; 27<\/span><\/strong>; <strong><span class=\"has-inline-color has-vivid-green-cyan-color\">81<\/span><\/strong>} teremos que:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"145\" height=\"98\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17125225\/image-254.png\" alt=\"\" class=\"wp-image-527880\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-resumo-estrategico\">Resumo Estrat\u00e9gico<\/h2>\n\n\n\n<p>Com objetivo de facilitar sua revis\u00e3o, vejamos uma tabela com as f\u00f3rmulas que <strong>resolvem a grande maioria dos exerc\u00edcios<\/strong> sobre Progress\u00e3o Aritm\u00e9tica e Progress\u00e3o Geom\u00e9trica.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"567\" height=\"349\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17125328\/image-255.png\" alt=\"Resumo PA e PG\" class=\"wp-image-527882\" srcset=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17125328\/image-255.png 567w, https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17125328\/image-255-300x185.png 300w\" sizes=\"auto, (max-width: 567px) 100vw, 567px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-venda-na-pratica\">Venda na pr\u00e1tica<\/h2>\n\n\n\n<p>Agora vamos resolver duas quest\u00f5es de PA e duas de PG para sedimentar o que vimos at\u00e9 aqui.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-exercicios-pa\">Exerc\u00edcios PA<\/h3>\n\n\n\n<p><strong>CESPE\/2019 &#8211; T\u00e9cnico (TJ PR)<\/strong><\/p>\n\n\n\n<p><strong>O protocolo de determinado tribunal associa, a cada dia, a ordem de chegada dos processos aos termos de uma progress\u00e3o aritm\u00e9tica de raz\u00e3o 2: a cada dia, o primeiro processo que chega recebe o n\u00famero 3, o segundo, o n\u00famero 5, e assim sucessivamente. Se, em determinado dia, o \u00faltimo processo que chegou ao protocolo recebeu o n\u00famero 69, ent\u00e3o, nesse dia, foram protocolados<\/strong><\/p>\n\n\n\n<p><strong>&nbsp;a) 23 processos.<\/strong><\/p>\n\n\n\n<p><strong>b) 33 processos.<\/strong><\/p>\n\n\n\n<p><strong>c) 34 processos.<\/strong><\/p>\n\n\n\n<p><strong>d) 66 processos.<\/strong><\/p>\n\n\n\n<p><strong>e) 67 processos.<\/strong><\/p>\n\n\n\n<p>Primeiro destacaremos as informa\u00e7\u00f5es importantes do enunciado:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>r = <\/strong>progress\u00e3o aritm\u00e9tica de raz\u00e3o <strong>2<\/strong><\/li><li><strong>a1<\/strong> = <img loading=\"lazy\" decoding=\"async\" width=\"31\" height=\"17\" src=\"\"\/>o primeiro processo que chega recebe o n\u00famero<strong> 3<\/strong><\/li><li><strong>an<\/strong> = <img loading=\"lazy\" decoding=\"async\" width=\"31\" height=\"17\" src=\"\"\/>o \u00faltimo processo que chegou ao protocolo recebeu o n\u00famero <strong>69<\/strong><\/li><\/ul>\n\n\n\n<p>Assim,<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"190\" height=\"237\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17131311\/image-267.png\" alt=\"\" class=\"wp-image-527909\"\/><\/figure>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">Alternativa C<\/span><\/strong><\/p>\n\n\n\n<p><strong>VUNESP\/2020 &#8211; Cabo (PMSP)<\/strong><\/p>\n\n\n\n<p><strong>O percurso de um treinamento de corrida \u00e9 composto por 5 etapas com dist\u00e2ncias diferentes em cada uma delas. Uma nova etapa sempre tem 100 metros a mais que a etapa anterior. Sabendo que a quarta etapa do treinamento \u00e9 percorrer 1 200 metros, a dist\u00e2ncia total do percurso \u00e9 igual a<\/strong><\/p>\n\n\n\n<p><strong>a) 6 100 metros.<\/strong><\/p>\n\n\n\n<p><strong>b) 5 900 metros.<\/strong><\/p>\n\n\n\n<p><strong>c) 5 700 metros.<\/strong><\/p>\n\n\n\n<p><strong>d) 5 500 metros.<\/strong><\/p>\n\n\n\n<p>Novamente vamos separar as informa\u00e7\u00f5es importantes:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>r <\/strong>= Nova etapa sempre tem <strong>100 metros<\/strong> a mais que a etapa anterior<\/li><li><strong>a4<\/strong> = Quarta etapa do treinamento \u00e9 percorrer <strong>1.200 metros<\/strong><\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"190\" height=\"176\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17125742\/image-258.png\" alt=\"\" class=\"wp-image-527889\"\/><\/figure>\n\n\n\n<p>A dist\u00e2ncia total do percurso \u00e9 igual a somat\u00f3ria dos termos da PA, logo:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"188\" height=\"296\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17125832\/image-259.png\" alt=\"\" class=\"wp-image-527890\"\/><\/figure>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">Alternativa D<\/span><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-exercicios-pg\">Exerc\u00edcios PG<\/h3>\n\n\n\n<p><strong>CESPE\/2020 &#8211; Analista (TJ PA)<\/strong><\/p>\n\n\n\n<p><strong>No dia 1.\u00ba de janeiro de 2019, uma nova secretaria foi criada em certo tribunal, a fim de receber todos os processos a serem protocolados nessa institui\u00e7\u00e3o. Durante o m\u00eas de janeiro de 2019, 10 processos foram protocolados nessa secretaria; a partir de ent\u00e3o, a quantidade mensal de processos protocolados na secretaria durante esse ano formou uma progress\u00e3o geom\u00e9trica de raz\u00e3o igual a 2.<\/strong><\/p>\n\n\n\n<p><strong>Nessa situa\u00e7\u00e3o hipot\u00e9tica, a quantidade de processos protocolados nessa secretaria durante os meses de junho e julho de 2019 foi igual a<\/strong><\/p>\n\n\n\n<p><strong>a) 320.<\/strong><\/p>\n\n\n\n<p><strong>b) 480.<\/strong><\/p>\n\n\n\n<p><strong>c) 640.<\/strong><\/p>\n\n\n\n<p><strong>d) 960.<\/strong><\/p>\n\n\n\n<p><strong>e) 1.270.<\/strong><\/p>\n\n\n\n<p>Dados da quest\u00e3o:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><img loading=\"lazy\" decoding=\"async\" width=\"31\" height=\"17\" src=\"\"\/><strong>a1<\/strong>= Durante o m\u00eas de janeiro de 2019, <strong>10 processos<\/strong> foram protocolados<\/li><li><strong>q<\/strong> = progress\u00e3o geom\u00e9trica de raz\u00e3o igual a<strong> 2<\/strong><\/li><\/ul>\n\n\n\n<p>Logo, a quantidade do m\u00eas junho de junho&nbsp;ser\u00e1:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"149\" height=\"183\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17130329\/image-260.png\" alt=\"\" class=\"wp-image-527895\"\/><\/figure>\n\n\n\n<p>Agora para encontrar o m\u00eas de julho basta multiplicar a quantidade de junho pela raz\u00e3o (lembre-se da defini\u00e7\u00e3o!).<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"125\" height=\"72\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17130404\/image-261.png\" alt=\"\" class=\"wp-image-527897\"\/><\/figure>\n\n\n\n<p>Assim, a quantidade de processos protocolados nessa secretaria durante os meses de junho e julho de 2019 foi igual a<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"138\" height=\"74\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17130427\/image-262.png\" alt=\"\" class=\"wp-image-527898\"\/><\/figure>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">Alternativa D<\/span><\/strong><\/p>\n\n\n\n<p><strong>&nbsp;FCC\/2019 &#8211; Analista (TRF 3\u00aa)<\/strong><\/p>\n\n\n\n<p><strong>Um carro percorreu 3.000 km. A cada dia de viagem, a partir do primeiro, ele dobrou a dist\u00e2ncia percorrida no dia anterior. Se ele finalizou a viagem em quatro dias, a dist\u00e2ncia percorrida, em quil\u00f4metros, no primeiro dia foi de<\/strong><\/p>\n\n\n\n<p><strong>a) 100.<\/strong><\/p>\n\n\n\n<p><strong>b) 200.<\/strong><\/p>\n\n\n\n<p><strong>c) 150.<\/strong><\/p>\n\n\n\n<p><strong>d) 250.<\/strong><\/p>\n\n\n\n<p><strong>e) 300.<\/strong><\/p>\n\n\n\n<p>Para finalizar, uma quest\u00e3o que aborda a soma dos termos de uma PG.<\/p>\n\n\n\n<p>Dados:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"128\" height=\"73\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17130602\/image-263.png\" alt=\"\" class=\"wp-image-527900\"\/><\/figure>\n\n\n\n<p>O primeiro ponto a se notar \u00e9 que a raz\u00e3o n\u00e3o est\u00e1 no intervalo de -1 e 1 e que se trata de uma soma finita de valores, ent\u00e3o:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"137\" height=\"68\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17130724\/image-264.png\" alt=\"\" class=\"wp-image-527902\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-amber-color\"><strong>Obs.<\/strong> <\/mark>Aten\u00e7\u00e3o, pois a somat\u00f3ria dos termos da PG, como vimos, tem duas f\u00f3rmulas poss\u00edveis, uma para termos infinitos e outra para termos finitos.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"180\" height=\"236\" src=\"https:\/\/dhg1h5j42swfq.cloudfront.net\/2020\/06\/17131219\/image-266.png\" alt=\"\" class=\"wp-image-527908\"\/><\/figure>\n\n\n\n<p><strong><span class=\"has-inline-color has-vivid-cyan-blue-color\">Alternativa B<\/span><\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-consideracoes-finais\">Considera\u00e7\u00f5es Finais<\/h2>\n\n\n\n<p>Caso tenha restado alguma d\u00favida ou queira aprofundar o conhecimento no tema, aconselho o curso b\u00e1sico de matem\u00e1tica nas aulas 22 e 23.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><a href=\"https:\/\/www.estrategiaconcursos.com.br\/curso\/curso-basico-de-matematica-para-concursos-com-videoaulas-2020\/\">Curso de matem\u00e1tica<\/a><\/p>\n\n\n\n<p>Por hoje \u00e9 s\u00f3, espero que o artigo tenha colaborado com seus estudos sobre Progress\u00e3o Aritm\u00e9tica e Progress\u00e3o Geom\u00e9trica.<\/p>\n\n\n\n<p>At\u00e9 a pr\u00f3xima;<\/p>\n\n\n\n<p>Leonardo Passarin<\/p>\n\n\n\n<p><strong>Instagram:<\/strong> @leonardompassarin<\/p>\n\n\n\n<p class=\"has-text-align-center has-medium-font-size\"><em><strong>Assinatura Anual Ilimitada*<\/strong><\/em><\/p>\n\n\n\n<p class=\"has-small-font-size\"><em>Prepare-se com o melhor material e com quem mais aprova em Concursos P\u00fablicos em todo o pa\u00eds. Assine agora a nossa Assinatura Anual e tenha acesso ilimitado* a todos os nossos cursos.<\/em><\/p>\n\n\n\n<p class=\"has-text-align-center has-luminous-vivid-amber-background-color has-background has-medium-font-size\"><strong>ASSINE AGORA<\/strong>&nbsp;\u2013&nbsp;<a rel=\"noreferrer noopener\" href=\"https:\/\/www.estrategiaconcursos.com.br\/assinaturas\" target=\"_blank\"><strong>Assinatura Ilimitada<\/strong><\/a><\/p>\n\n\n\n<p><em><strong>Fique por dentro de todos os concursos<\/strong><\/em>:<\/p>\n\n\n\n<p class=\"has-text-align-center has-very-dark-gray-color has-vivid-cyan-blue-background-color has-text-color has-background has-medium-font-size\"><a rel=\"noreferrer noopener\" href=\"https:\/\/www.estrategiaconcursos.com.br\/blog\/concursos-abertos\/\" target=\"_blank\"><strong>Concursos abertos<\/strong><\/a><\/p>\n\n\n\n<p class=\"has-text-align-center has-very-dark-gray-color has-vivid-cyan-blue-background-color has-text-color has-background has-medium-font-size\"><a rel=\"noreferrer noopener\" href=\"https:\/\/www.estrategiaconcursos.com.br\/blog\/concursos-2020\/\" target=\"_blank\"><strong>Concursos 2020<\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ol\u00e1, pessoal, no artigo de hoje trataremos sobre as aplica\u00e7\u00f5es de Progress\u00e3o Aritm\u00e9tica e Progress\u00e3o Geom\u00e9trica nas provas de concurso. Vamos l\u00e1? Defini\u00e7\u00e3o Progress\u00e3o nada mais \u00e9 que o desenvolvimento de uma sequ\u00eancia que segue uma lei de forma\u00e7\u00e3o (uma f\u00f3rmula matem\u00e1tica). No nosso cotidiano estamos diante de v\u00e1rias sequ\u00eancias, tais como: Anivers\u00e1rios: (20 anos, [&hellip;]<\/p>\n","protected":false},"author":741,"featured_media":531099,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"post_tipo":"article","footnotes":""},"categories":[2313,1,3766,1983,2017,1988,1984],"tags":[],"tax_estado":[],"class_list":["post-527758","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-carreiras-bancarias","category-cursos-e-concursos","category-educacional","category-fiscal-estadual","category-concursos-legislativos","category-policia","category-tribunais"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.2 (Yoast SEO v27.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Progress\u00e3o Aritm\u00e9tica e Progress\u00e3o Geom\u00e9trica \u2013 Aplica\u00e7\u00e3o em concursos<\/title>\n<meta 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