{"id":95554,"date":"2017-01-19T12:32:49","date_gmt":"2017-01-19T14:32:49","guid":{"rendered":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/?p=95554"},"modified":"2026-06-01T15:14:40","modified_gmt":"2026-06-01T17:14:40","slug":"progressao-geometrica","status":"publish","type":"post","link":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/matematica\/progressao-geometrica","title":{"rendered":"Progress\u00e3o Geom\u00e9trica"},"content":{"rendered":"<p>Uma<strong>\u00a0Progress\u00e3o geom\u00e9trica (PG)<\/strong>\u00a0\u00e9 uma sequ\u00eancia de n\u00fameros que crescem ou decrescem segundo um fator multiplicativo. Ela pode ser representada pela raz\u00e3o\u00a0<strong>q<\/strong>, o termo inicial\u00a0<em>a<sub>1<\/sub><\/em>.<\/p>\n<p>Uma\u00a0<b>progress\u00e3o geom\u00e9trica (PG)<\/b>\u00a0\u00e9 definida como uma sequ\u00eancia onde cada termo \u00e9 obtido multiplicando-se o termo anterior por um n\u00famero fixo chamado raz\u00e3o. Esse tipo de progress\u00e3o \u00e9 comumente usado para representar crescimento ou decaimento exponencial.<\/p>\n<p>Uma\u00a0<b>progress\u00e3o geom\u00e9trica<\/b>\u00a0\u00e9 um tipo especial de progress\u00e3o\/sequ\u00eancia. Para que uma sequ\u00eancia de n\u00fameros seja considerada uma progress\u00e3o geom\u00e9trica, todos os seus elementos devem ser multiplicados por uma raz\u00e3o comum. O n\u00famero anterior na sequ\u00eancia, multiplicado pela raz\u00e3o comum, fornecer\u00e1 o pr\u00f3ximo n\u00famero na progress\u00e3o geom\u00e9trica.<\/p>\n<p><b>O n-\u00e9simo termo da\u00a0<\/b><strong>Progress\u00e3o geom\u00e9trica (PG)<\/strong><b>\u00a0\u00e9 dado por:<\/b><\/p>\n<p><img decoding=\"async\" class=\"alignnone wp-image-127335 size-full\" title=\"F\u00f3rmula da Progress\u00e3o Geom\u00e9trica - PG\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-1.jpg\" alt=\"F\u00f3rmula da Progress\u00e3o Geom\u00e9trica - PG\" width=\"162\" height=\"35\" \/><\/p>\n<p>Dizemos que a PG \u00e9 crescente quando a raz\u00e3o <strong>q <\/strong>\u00e9 tal que <strong>q<\/strong> &gt; 1 e decrescente quando\u00a00 &lt;<strong> q<\/strong> &lt;1. Nomeamos oscilantes as PGs que possuem <strong>q\u00a0<\/strong>&lt; 0.<\/p>\n<p>Uma propriedade bastante relevante da PG \u00e9 que podemos obter os termos imediatamente antes e depois de um termo conhecido a partir da sua raz\u00e3o:<\/p>\n<p class=\"imagem\"><img decoding=\"async\" class=\"alignnone size-full wp-image-127334\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-2.jpg\" alt=\"F\u00f3rmula da Progress\u00e3o Geom\u00e9trica - PG\" width=\"153\" height=\"75\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-2.jpg 153w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-2-150x75.jpg 150w\" sizes=\"(max-width: 153px) 100vw, 153px\" \/><\/p>\n<p><strong>Vejamos alguns exemplos:<\/strong><\/p>\n<p><strong>Exemplo) <\/strong>Para uma PG de raz\u00e3o q = \u00bd, iniciando em , quais s\u00e3o os primeiros 5 termos?<\/p>\n<p>Os primeiros cinco termos s\u00e3o: 1, 0.5 , 0.25 , 0.125 , 0.625\u00a0ou ent\u00e3o: 1, \u00bd, \u00bc, \u215b, 1\/16<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-127336\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-1.jpg\" alt=\"progressao-geometrica-exemplo-1\" width=\"213\" height=\"151\" \/><\/p>\n<p>Dizemos que esta \u00e9 uma PG decrescente.<\/p>\n<p><strong>Exemplo)\u00a0<\/strong>Para uma PG de raz\u00e3o q = 3 cujo terceiro termo \u00e9 27, quais s\u00e3o os termos que o antecede e sucede?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127337\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-2.jpg\" alt=\"progressao-geometrica-exemplo-2\" width=\"186\" height=\"78\" \/><\/p>\n<p><strong>Soma da PG finita<\/strong><\/p>\n<p>Para obter a soma dos primeiros n termos da PG, fazemos<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-127338 size-full\" title=\"F\u00f3rmula da Soma de uma PG Finita\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-3.jpg\" alt=\"F\u00f3rmula da Soma de uma PG Finita\" width=\"155\" height=\"49\" \/><\/p>\n<p><strong>Exemplo)<\/strong>\u00a0Calcule os primeiros 7 termos de uma PG de raz\u00e3o 2, cujo termo inicial \u00e9 1.<\/p>\n<p>Utilizando a f\u00f3rmula<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127339\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-3.jpg\" alt=\"progressao-geometrica-exemplo-3\" width=\"347\" height=\"51\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-3.jpg 347w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-3-300x44.jpg 300w\" sizes=\"(max-width: 347px) 100vw, 347px\" \/><\/p>\n<p>Vamos confirmar a partir da soma de cada termo:<\/p>\n<p>a<sub>1<\/sub> = 1<br \/>\na<sub>2<\/sub>\u00a0= 1 * 2\u00b9 = 2<br \/>\na<sub>3<\/sub>\u00a0= 1 * 2<sup>2<\/sup> =\u00a04<br \/>\na<sub>4<\/sub>\u00a0= 1 * 2<sup>3<\/sup>\u00a0=\u00a08<br \/>\na<sub>5<\/sub>\u00a0= 1 * 2<sup>4<\/sup> =\u00a016<br \/>\na<sub>6<\/sub>\u00a0= 1 * 2<sup>5<\/sup> =\u00a032<br \/>\na<sub>7<\/sub>\u00a0= 1 * 2<sup>6<\/sup> =\u00a064<br \/>\nS<sub>7<\/sub> = a<sub>1<\/sub> + a<sub>2<\/sub> + a<sub>3<\/sub> + a<sub>4<\/sub> + a<sub>5<\/sub> + a<sub>6<\/sub> + a<sub>7<\/sub> = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127<\/p>\n<p><strong>Exemplo)<\/strong>\u00a0Calcule os primeiros 3 termos de uma PG de raz\u00e3o 1\/3, cujo termo inicial \u00e9 3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127346\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-4.jpg\" alt=\"progressao-geometrica-exemplo-4\" width=\"473\" height=\"50\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-4.jpg 473w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-4-300x32.jpg 300w\" sizes=\"(max-width: 473px) 100vw, 473px\" \/><\/p>\n<p>Vamos confirmar a partir da soma de cada termo<\/p>\n<p>a<sub>1<\/sub> = 3<br \/>\na<sub>2<\/sub> = 3 * (1\/3)\u00b9 = 1<br \/>\na<sub>3<\/sub> = 3 * (1\/3)\u00b2 = 1\/3<br \/>\nS<sub>3<\/sub> = a<sub>1<\/sub> + a<sub>2<\/sub> + a<sub>3<\/sub> = 3 + 1 + 1\/3 = 9\/3 + 3\/3 + 1\/3 = 13\/3<\/p>\n<p><strong>Soma da PG infinita<\/strong><\/p>\n<p>Calculamos a soma de uma progress\u00e3o geom\u00e9trica infinita quando a sua raz\u00e3o \u00e9 do tipo 0 &lt; q &lt; 1, isto \u00e9, para uma PG decrescente, assim, podemos garantir que o resultado dela converge para um valor determinado.<\/p>\n<p>Vamos ver abaixo um argumento geom\u00e9trico da converg\u00eancia do termo infinito ao zero exemplificado pela PG decrescente iniciando no termo de valor 1 com raz\u00e3o q = \u00bd.<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127348\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/soma-progressao-geometrica-infinita.jpg\" alt=\"soma-progressao-geometrica-infinita\" width=\"433\" height=\"256\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/soma-progressao-geometrica-infinita.jpg 433w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/soma-progressao-geometrica-infinita-300x177.jpg 300w\" sizes=\"(max-width: 433px) 100vw, 433px\" \/><\/p>\n<p>Vamos indicar a soma da PG infinita como<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-127349 size-full\" title=\"F\u00f3rmula para soma de uma PG infinita\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/formula-soma-de-uma-pg-infinita.jpg\" alt=\"F\u00f3rmula para soma de uma PG infinita\" width=\"101\" height=\"46\" \/><\/p>\n<p>Calculando a soma da PG infinita para o exemplo acima (a<sub>1<\/sub> = q, q = 1\/2), temos que<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127350\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/formula-soma-de-uma-pg-infinita-2.jpg\" alt=\"formula-soma-de-uma-pg-infinita-2\" width=\"162\" height=\"38\" \/><\/p>\n<p class=\"normal\">Geometricamente, confirmamos este resultado.<\/p>\n<p class=\"normal\"><strong>Exemplo)<\/strong>\u00a0Qual \u00e9 a soma infinita da Progress\u00e3o Geom\u00e9trica iniciada em 100 de raz\u00e3o q = \u2153?<\/p>\n<p class=\"normal\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127351\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-geometrica-exemplo-5.jpg\" alt=\"progressao-geometrica-exemplo-5\" width=\"189\" height=\"37\" \/><\/p>\n<p class=\"normal\"><strong>Fonte:<\/strong>\u00a0Col\u00e9gio S\u00e7ao Francisco\/Luisa Boccardo Burini<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Uma\u00a0Progress\u00e3o geom\u00e9trica (PG)\u00a0\u00e9 uma sequ\u00eancia de n\u00fameros que crescem ou decrescem segundo um fator multiplicativo. Ela pode ser representada pela raz\u00e3o\u00a0q, o termo inicial\u00a0a1. Uma\u00a0progress\u00e3o geom\u00e9trica (PG)\u00a0\u00e9 definida como uma sequ\u00eancia onde cada termo \u00e9 obtido multiplicando-se o termo anterior por um n\u00famero fixo chamado raz\u00e3o. Esse tipo de progress\u00e3o \u00e9 comumente usado para representar [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[414],"tags":[],"class_list":["post-95554","post","type-post","status-publish","format-standard","hentry","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Progress\u00e3o Geom\u00e9trica, Sequ\u00eancia, N\u00fameros Progress\u00e3o Geom\u00e9trica<\/title>\n<meta name=\"description\" content=\"Progress\u00e3o Geom\u00e9trica, Sequ\u00eancia, N\u00fameros, Decrescente, Raz\u00e3o, Soma, Infinita, Termos, F\u00f3rmula, Exemplo, Progress\u00e3o Geom\u00e9trica\" \/>\n<meta name=\"robots\" content=\"noindex, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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