{"id":484,"date":"2023-11-16T15:20:09","date_gmt":"2023-11-16T15:20:09","guid":{"rendered":"https:\/\/www.ginasiosdavinci.com\/blog\/?p=484"},"modified":"2023-11-16T19:11:54","modified_gmt":"2023-11-16T19:11:54","slug":"curiosidades-da-matematica-exemplo-de-progressao-simples-e-surpreendente-de-uma-progressao-geometrica-a-resposta-e-surpreendente","status":"publish","type":"post","link":"https:\/\/www.ginasiosdavinci.com\/blog\/2023\/11\/16\/curiosidades-da-matematica-exemplo-de-progressao-simples-e-surpreendente-de-uma-progressao-geometrica-a-resposta-e-surpreendente\/","title":{"rendered":"Curiosidades da matem\u00e1tica: Exemplo de progress\u00e3o geom\u00e9trica simples. A resposta \u00e9 surpreendente."},"content":{"rendered":"\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/www.ginasiosdavinci.com\/blog\/wp-content\/uploads\/2023\/11\/progressao-geometrica-1.webp\" alt=\"\" class=\"wp-image-489\" srcset=\"https:\/\/www.ginasiosdavinci.com\/blog\/wp-content\/uploads\/2023\/11\/progressao-geometrica-1.webp 1024w, https:\/\/www.ginasiosdavinci.com\/blog\/wp-content\/uploads\/2023\/11\/progressao-geometrica-1-300x300.webp 300w, https:\/\/www.ginasiosdavinci.com\/blog\/wp-content\/uploads\/2023\/11\/progressao-geometrica-1-150x150.webp 150w, https:\/\/www.ginasiosdavinci.com\/blog\/wp-content\/uploads\/2023\/11\/progressao-geometrica-1-768x768.webp 768w, https:\/\/www.ginasiosdavinci.com\/blog\/wp-content\/uploads\/2023\/11\/progressao-geometrica-1-100x100.webp 100w, https:\/\/www.ginasiosdavinci.com\/blog\/wp-content\/uploads\/2023\/11\/progressao-geometrica-1-116x116.webp 116w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>E se pedisses aos teus pais que te dessem uma mesada nos seguintes termos: Hoje davam-te 1 c\u00eantimo, amanh\u00e3 2, depois 4, 8, 16, 32, por a\u00ed fora. Todos os dias davam-te o dobro do dia anterior. <br><\/p>\n\n\n\n<p>Ao fim de 10 dias terias 10,24\u20ac. Sabes quanto terias ao fim do m\u00eas? A resposta \u00e9 surpreendente, prepara-te para ficar de boca aberta!<\/p>\n\n\n\n<p>Terias mais de 10 milh\u00f5es de Euros! Verdade, \u00e9 o que se chama uma progress\u00e3o geom\u00e9trica. Aqui ficam os c\u00e1lculos e a justifica\u00e7\u00e3o:<\/p>\n\n\n\n<p>Este problema \u00e9 um exemplo cl\u00e1ssico de uma progress\u00e3o geom\u00e9trica, onde cada termo \u00e9 o dobro do termo anterior. Por exemplo, para calcular o total que terias ao fim de 10 dias, podes usar a f\u00f3rmula da soma de n termos de uma progress\u00e3o geom\u00e9trica ou fazer uma tabela e somar os valores. <\/p>\n\n\n\n<p>A tabela \u00e9:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Dia<\/strong><\/td><td><strong>Valor (\u20ac)<\/strong><\/td><\/tr><\/thead><tbody><tr><td>1<\/td><td>0.01<\/td><\/tr><tr><td>2<\/td><td>0.02<\/td><\/tr><tr><td>3<\/td><td>0.04<\/td><\/tr><tr><td>4<\/td><td>0.08<\/td><\/tr><tr><td>5<\/td><td>0.16<\/td><\/tr><tr><td>6<\/td><td>0.32<\/td><\/tr><tr><td>7<\/td><td>0.64<\/td><\/tr><tr><td>8<\/td><td>1.28<\/td><\/tr><tr><td>9<\/td><td>2.56<\/td><\/tr><tr><td>10<\/td><td>5.12<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Se somares os valores recebidos todos os dias obt\u00e9ns = 10,24\u20ac<\/p>\n\n\n\n<p>Mais mais f\u00e1cil \u00e9 utilizar a f\u00f3rmula para calcular a soma de n termos numa progress\u00e3o geom\u00e9trica, que \u00e9 Sn\u200b=a1\u200b\u00d7(r<sup>n<\/sup>\u22121\u200b)\/(r\u22121)<\/p>\n\n\n\n<p>onde:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sn\u200b \u00e9 a soma dos n termos<\/li>\n\n\n\n<li>a1\u200b \u00e9 o primeiro termo (neste caso 1 c\u00eantimo (0,01\u20ac), que \u00e9 o que os teus pais te d\u00e3o no 1\u00ba dia)<\/li>\n\n\n\n<li>r \u00e9 a raz\u00e3o (neste caso, 2, porque cada termo \u00e9 o dobro do anterior)<\/li>\n\n\n\n<li>n \u00e9 o n\u00famero de termos (neste caso 10, porque s\u00e3o 10 dias)<\/li>\n<\/ul>\n\n\n\n<p>Assim, substituindo os valores na f\u00f3rmula, obt\u00e9ns os mesmos 10,24\u20ac &nbsp;<\/p>\n\n\n\n<p>E ao fim do m\u00eas? &nbsp;<\/p>\n\n\n\n<p>Usando a mesma f\u00f3rmula para calcular a soma dos termos de uma progress\u00e3o geom\u00e9trica para 31 dias, terias ao fim do m\u00eas mais de 10 milh\u00f5es de euros, terias 10.737,418,23 euros!<\/p>\n\n\n\n<p>Eis o comprovativo em forma de tabela:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Dia<\/strong><\/td><td><strong>Valor do Dia (\u20ac)<\/strong><\/td><td><strong>Acumulado (\u20ac)<\/strong><\/td><\/tr><\/thead><tbody><tr><td>1<\/td><td>0.01<\/td><td>0.01<\/td><\/tr><tr><td>2<\/td><td>0.02<\/td><td>0.03<\/td><\/tr><tr><td>3<\/td><td>0.04<\/td><td>0.07<\/td><\/tr><tr><td>4<\/td><td>0.08<\/td><td>0.15<\/td><\/tr><tr><td>5<\/td><td>0.16<\/td><td>0.31<\/td><\/tr><tr><td>6<\/td><td>0.32<\/td><td>0.63<\/td><\/tr><tr><td>7<\/td><td>0.64<\/td><td>1.27<\/td><\/tr><tr><td>8<\/td><td>1.28<\/td><td>2.55<\/td><\/tr><tr><td>9<\/td><td>2.56<\/td><td>5.11<\/td><\/tr><tr><td>10<\/td><td>5.12<\/td><td>10.24<\/td><\/tr><tr><td>11<\/td><td>10.24<\/td><td>20.47<\/td><\/tr><tr><td>12<\/td><td>20.48<\/td><td>40.95<\/td><\/tr><tr><td>13<\/td><td>40.96<\/td><td>81.91<\/td><\/tr><tr><td>14<\/td><td>81.92<\/td><td>163.83<\/td><\/tr><tr><td>15<\/td><td>163.84<\/td><td>327.67<\/td><\/tr><tr><td>16<\/td><td>327.68<\/td><td>655.35<\/td><\/tr><tr><td>17<\/td><td>655.36<\/td><td>1310.71<\/td><\/tr><tr><td>18<\/td><td>1310.72<\/td><td>2621.43<\/td><\/tr><tr><td>19<\/td><td>2621.44<\/td><td>5242.87<\/td><\/tr><tr><td>20<\/td><td>5242.88<\/td><td>10485.75<\/td><\/tr><tr><td>21<\/td><td>10485.76<\/td><td>20971.51<\/td><\/tr><tr><td>22<\/td><td>20971.52<\/td><td>41943.03<\/td><\/tr><tr><td>23<\/td><td>41943.04<\/td><td>83886.07<\/td><\/tr><tr><td>24<\/td><td>83886.08<\/td><td>167772.15<\/td><\/tr><tr><td>25<\/td><td>167772.16<\/td><td>335544.31<\/td><\/tr><tr><td>26<\/td><td>335544.32<\/td><td>671088.63<\/td><\/tr><tr><td>27<\/td><td>671088.64<\/td><td>1342177.27<\/td><\/tr><tr><td>28<\/td><td>1342177.28<\/td><td>2684354.55<\/td><\/tr><tr><td>29<\/td><td>2684354.56<\/td><td>5368709.11<\/td><\/tr><tr><td>30<\/td><td>5368709.12<\/td><td><strong>10.737.418,23<\/strong><strong><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Gin\u00e1sios da Educa\u00e7\u00e3o Da Vinci, onde h\u00e1 sempre mais para aprender<\/strong>.<\/p>\n\n\n\n<p>Procura-nos <a href=\"https:\/\/www.ginasiosdavinci.com\/\"><strong>https:\/\/www.ginasiosdavinci.com\/<\/strong><\/a><\/p>\n\n\n\n<p><\/p>\n<div class=\"likebtn_container\" style=\"\"><!-- LikeBtn.com BEGIN --><span class=\"likebtn-wrapper\"  data-identifier=\"post_484\"  data-lang=\"pt\"  data-style=\"\"  data-unlike_allowed=\"\"  data-show_copyright=\"\"  data-item_url=\"https:\/\/www.ginasiosdavinci.com\/blog\/2023\/11\/16\/curiosidades-da-matematica-exemplo-de-progressao-simples-e-surpreendente-de-uma-progressao-geometrica-a-resposta-e-surpreendente\/\"  data-item_title=\"Curiosidades da matem\u00e1tica: Exemplo de progress\u00e3o geom\u00e9trica simples. A resposta \u00e9 surpreendente.\"  data-item_image=\"https:\/\/www.ginasiosdavinci.com\/blog\/wp-content\/uploads\/2023\/11\/progressao-geometrica.webp\"  data-item_date=\"2023-11-16T15:20:09+00:00\"  data-engine=\"WordPress\"  data-plugin_v=\"2.6.48\"  data-event_handler=\"likebtn_eh\" ><\/span><!-- LikeBtn.com END --><\/div>","protected":false},"excerpt":{"rendered":"<p>E se pedisses aos teus pais que te dessem uma mesada nos seguintes termos: Hoje davam-te 1 c\u00eantimo, amanh\u00e3 2, depois 4, 8, 16, 32, por a\u00ed fora. Todos os dias davam-te o dobro do dia anterior. Ao fim de 10 dias terias 10,24\u20ac. Sabes quanto terias ao fim do m\u00eas? A resposta \u00e9 surpreendente, [&hellip;]<\/p>\n<div class=\"likebtn_container\" style=\"\"><!-- LikeBtn.com BEGIN --><span class=\"likebtn-wrapper\"  data-identifier=\"post_484\"  data-lang=\"pt\"  data-style=\"\"  data-unlike_allowed=\"\"  data-show_copyright=\"\"  data-item_url=\"https:\/\/www.ginasiosdavinci.com\/blog\/2023\/11\/16\/curiosidades-da-matematica-exemplo-de-progressao-simples-e-surpreendente-de-uma-progressao-geometrica-a-resposta-e-surpreendente\/\"  data-item_title=\"Curiosidades da matem\u00e1tica: Exemplo de progress\u00e3o geom\u00e9trica simples. 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