{"id":95497,"date":"2017-01-19T09:29:39","date_gmt":"2017-01-19T11:29:39","guid":{"rendered":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/?p=95497"},"modified":"2026-06-01T15:13:06","modified_gmt":"2026-06-01T17:13:06","slug":"progressao-aritmetica","status":"publish","type":"post","link":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/matematica\/progressao-aritmetica","title":{"rendered":"Progress\u00e3o Aritm\u00e9tica"},"content":{"rendered":"<p><b>Progress\u00e3o Aritm\u00e9tica &#8211; O que \u00e9<\/b><\/p>\n<p>Uma progress\u00e3o \u00e9 uma sucess\u00e3o de n\u00fameros. Nessa sequ\u00eancia, existe um n\u00famero que a inicia, chamado de constante\u00a0 e para obter o pr\u00f3ximo n\u00famero, temos uma raz\u00e3o r \u2208 R.<\/p>\n<p><b>A\u00a0<\/b><strong>P<\/strong><b>rogress\u00e3o\u00a0<\/b><strong>A<\/strong><b>ritm\u00e9tica, ou\u00a0<\/b><strong>PA<\/strong><b>, \u00e9 uma sequ\u00eancia de n\u00fameros em que o n-\u00e9simo termo\u00a0<em>a<sub>i<\/sub><\/em>\u00a0pode ser descrito como:<\/b><\/p>\n<p><em>a<sub>n<\/sub>\u00a0= r + a\u00a0<sub>n-1\u00a0<\/sub><\/em>tal que\u00a0<em>a<sub>n<\/sub>, r, a\u00a0<sub>n-1<\/sub>\u00a0<\/em>\u2208 R<\/p>\n<p>Para descobrir qual \u00e9 a raz\u00e3o, basta subtrairmos dois termos sequenciais:<\/p>\n<p><em>a<sub>n<\/sub>\u00a0= r + a\u00a0<sub>n-1\u00a0<\/sub><\/em>\u21d2 r =\u00a0<em>a<sub>n \u2013\u00a0<\/sub>a<sub>\u00a0n-1<\/sub><\/em><\/p>\n<p><b>A forma mais generalizada de escrevermos uma PA \u00e9 utilizando tamb\u00e9m o seu primeiro termo, a constante:<\/b><\/p>\n<p><em>a<sub>n<\/sub>\u00a0=\u00a0a<sub>1<\/sub>\u00a0+ (n \u2013 1) * r,\u00a0<\/em>para\u00a0n\u00a0\u2208\u00a0<em>N<\/em><\/p>\n<p><strong>Propriedade de uma PA<\/strong><\/p>\n<p><b>Para descobrir um termo n da PA sendo que j\u00e1 sabemos o termo m dela e sua raz\u00e3o, basta equacionar a forma geral para ambos os termos\u00a0<em>a<sub>n<\/sub><\/em>\u00a0e\u00a0<em>a<sub>m<\/sub><\/em>:<\/b><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-126966\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-1.png\" alt=\"progressao-aritmetica-1\" width=\"280\" height=\"78\" \/><\/p>\n<p>Agora, para isolar o termo desconhecido, iremos subtrair as equa\u00e7\u00f5es (1) &#8211; (2)<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-126967\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-2.png\" alt=\"progressao-aritmetica-2\" width=\"327\" height=\"146\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-2.png 327w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-2-300x134.png 300w\" sizes=\"(max-width: 327px) 100vw, 327px\" \/><\/p>\n<p>Ent\u00e3o teremos a forma geral:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-126969\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-3.png\" alt=\"progressao-aritmetica-3\" width=\"306\" height=\"35\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-3.png 306w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-3-300x34.png 300w\" sizes=\"(max-width: 306px) 100vw, 306px\" \/><\/p>\n<p>Utilizando a equa\u00e7\u00e3o descoberta acima, observamos que um termo pode ser obtido atrav\u00e9s da m\u00e9dia do posterior e do anterior:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-126971\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-4.png\" alt=\"progressao-aritmetica-4\" width=\"285\" height=\"135\" \/><\/p>\n<p>ent\u00e3o<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-126972\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-5.png\" alt=\"progressao-aritmetica-5\" width=\"355\" height=\"36\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-5.png 355w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-5-300x30.png 300w\" sizes=\"(max-width: 355px) 100vw, 355px\" \/><\/p>\n<p><strong>Soma de termos finitos de uma PA<\/strong><\/p>\n<p>Queremos descobrir qual \u00e9 a soma S p,q\u00a0dos termos de uma progress\u00e3o aritm\u00e9tica entre ap e aq.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-126973\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-6.png\" alt=\"progressao-aritmetica-6\" width=\"284\" height=\"37\" \/><\/p>\n<p>Primeiramente, vamos avaliar que a soma dos termos equidistantes dos extremos da Progress\u00e3o Aritm\u00e9tica s\u00e3o equivalentes a soma dos termos extremos, isto \u00e9, em uma soma S<sub>1, n<\/sub>\u00a0vamos ver as seguintes somas:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-126974\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-7.png\" alt=\"progressao-aritmetica-7\" width=\"570\" height=\"69\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-7.png 570w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-7-300x36.png 300w\" sizes=\"(max-width: 570px) 100vw, 570px\" \/><\/p>\n<p>Assim em diante!<\/p>\n<p>Logo, pela f\u00f3rmula do termo mais generalizada, podemos representar a soma S<sub>p,q<\/sub> como:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-126975\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-8.png\" alt=\"progressao-aritmetica-8\" width=\"500\" height=\"114\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-8.png 500w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-8-300x68.png 300w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><\/p>\n<p>Ent\u00e3o:<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-126976\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-9.png\" alt=\"progressao-aritmetica-9\" width=\"320\" height=\"67\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-9.png 320w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/progressao-aritmetica-9-300x63.png 300w\" sizes=\"(max-width: 320px) 100vw, 320px\" \/><\/p>\n<p><strong>Fonte:<\/strong>\u00a0Col\u00e9gio S\u00e3o Francisco\/Luisa Boccardo Burini<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Progress\u00e3o Aritm\u00e9tica &#8211; O que \u00e9 Uma progress\u00e3o \u00e9 uma sucess\u00e3o de n\u00fameros. Nessa sequ\u00eancia, existe um n\u00famero que a inicia, chamado de constante\u00a0 e para obter o pr\u00f3ximo n\u00famero, temos uma raz\u00e3o r \u2208 R. A\u00a0Progress\u00e3o\u00a0Aritm\u00e9tica, ou\u00a0PA, \u00e9 uma sequ\u00eancia de n\u00fameros em que o n-\u00e9simo termo\u00a0ai\u00a0pode ser descrito como: an\u00a0= r + a\u00a0n-1\u00a0tal [&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-95497","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 Aritm\u00e9tica, Sucess\u00e3o de N\u00fameros Progress\u00e3o Aritm\u00e9tica<\/title>\n<meta name=\"description\" content=\"Progress\u00e3o Aritm\u00e9tica, Sucess\u00e3o de N\u00fameros, Sequ\u00eancia, Constante, Raz\u00e3o, Soma, Termo, F\u00f3rmula, Propriedade, Progress\u00e3o 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