{"id":1874,"date":"2020-12-26T08:40:53","date_gmt":"2020-12-26T08:40:53","guid":{"rendered":"https:\/\/blogs.ua.es\/dimates\/?p=1874"},"modified":"2020-12-26T08:40:53","modified_gmt":"2020-12-26T08:40:53","slug":"solucion-a-suma-2020","status":"publish","type":"post","link":"https:\/\/blogs.ua.es\/dimates\/2020\/12\/26\/solucion-a-suma-2020\/","title":{"rendered":"Soluci\u00f3n a suma 2020"},"content":{"rendered":"<pre>Problema 0 del concurso Olitele 2020\r\nSe dirige a una edad de: 16-17 a\u00f1os<\/pre>\n<p>Calcula el valor de la expresi\u00f3n:<\/p>\n<p>2099\u00b2 \u2013 2098\u00b2 + 2097\u00b2 \u2013 2096\u00b2 + \u2026 + 2023\u00b2 \u2013 2022\u00b2 + 2021\u00b2 \u2013 2020\u00b2<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1872\" src=\"https:\/\/blogs.ua.es\/dimates\/files\/2020\/12\/174.Suma2020.png\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/blogs.ua.es\/dimates\/files\/2020\/12\/174.Suma2020.png 300w, https:\/\/blogs.ua.es\/dimates\/files\/2020\/12\/174.Suma2020-150x150.png 150w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\nSoluci\u00f3n:<br \/>\n<!--more--><br \/>\nEn este caso, tratamos de sumar restas de cuadrados. Puesto que s\u00b2 \u2013 t\u00b2 = (s + t)(s \u2013 t), y en este caso se trata de n\u00fameros consecutivos de la forma 2n + 1 y 2n (impar y par, respectivamente), podemos aplicar la f\u00f3rmula (2n + 1)\u00b2 \u2013 (2n)\u00b2 = 4n + 1.<\/p>\n<p>Es sencillo de comprobar, aunque sea de manera pr\u00e1ctica (probando en valores m\u00e1s peque\u00f1os), que las diferencias disminuyen de 4 en 4. La m\u00e1s grande es 4197 y la menor es 4041. En total, son 40 t\u00e9rminos, que forman una sucesi\u00f3n aritm\u00e9tica, cuya diferencia es 4.<\/p>\n<p>El primer t\u00e9rmino es 4041, y el t\u00e9rmino 40 ser\u00e1 4041 + 39\u00b74 = 4197, en efecto.<\/p>\n<p>Y sumar estos t\u00e9rminos en progresi\u00f3n aritm\u00e9tica, mediante la estrategia de sumarlos dos veces cambiando el orden para obtener sumas iguales, ser\u00e1 (4197 + 4041)\u00b740\/2 = 164760.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Problema 0 del concurso Olitele 2020 Se dirige a una edad de: 16-17 a\u00f1os Calcula el valor de la expresi\u00f3n: 2099\u00b2 \u2013 2098\u00b2 + 2097\u00b2 \u2013 2096\u00b2 + \u2026 + 2023\u00b2 \u2013 2022\u00b2 + 2021\u00b2 \u2013 2020\u00b2 Soluci\u00f3n:<\/p>\n","protected":false},"author":4267,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1738,2242014,2849,3303],"tags":[],"class_list":["post-1874","post","type-post","status-publish","format-standard","hentry","category-olimpiadas","category-olitele","category-problemas","category-soluciones"],"_links":{"self":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/1874","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/users\/4267"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/comments?post=1874"}],"version-history":[{"count":1,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/1874\/revisions"}],"predecessor-version":[{"id":1875,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/1874\/revisions\/1875"}],"wp:attachment":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/media?parent=1874"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/categories?post=1874"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/tags?post=1874"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}