{"id":772,"date":"2021-09-19T05:00:27","date_gmt":"2021-09-19T03:00:27","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=772"},"modified":"2021-09-15T11:18:10","modified_gmt":"2021-09-15T09:18:10","slug":"suma-de-progresion-aritmetica","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/suma-de-progresion-aritmetica\/","title":{"rendered":"Suma de progresi\u00f3n aritm\u00e9tica"},"content":{"rendered":"<p>Demostrar que la suma de los t\u00e9rminos de la progresi\u00f3n aritm\u00e9tica<\/p>\n<pre lang=\"text\">\n   a + (a + d) + (a + 2 \u00d7 d) + \u00b7\u00b7\u00b7 + (a + n \u00d7 d)\n<\/pre>\n<p>es (n + 1) \u00d7 (2 \u00d7 a + n \u00d7 d) \/ 2.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\nopen nat\n\nvariable  (n : \u2115)\nvariables (a d : \u211d)\n\n@[simp]\ndef sumaPA : \u211d \u2192 \u211d \u2192 \u2115 \u2192 \u211d\n| a d 0       := a\n| a d (n + 1) := sumaPA a d n + (a + (n + 1) * d)\n\nexample :\n  2 * sumaPA a d n = (n + 1) * (2 * a + n * d) :=\nsorry\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\r\nimport data.real.basic\r\nopen nat\r\n\r\nvariable  (n : \u2115)\r\nvariables (a d : \u211d)\r\n\r\nset_option pp.structure_projections false\r\n\r\n@[simp]\r\ndef sumaPA : \u211d \u2192 \u211d \u2192 \u2115 \u2192 \u211d\r\n| a d 0       := a\r\n| a d (n + 1) := sumaPA a d n + (a + (n + 1) * d)\r\n\r\nexample :\r\n  2 * sumaPA a d n = (n + 1) * (2 * a + n * d) :=\r\nbegin\r\n  induction n with n HI,\r\n  { simp, },\r\n  { calc 2 * sumaPA a d (succ n)\r\n         = 2 * (sumaPA a d n + (a + (n + 1) * d))\r\n           : rfl\r\n     ... = 2 * sumaPA a d n + 2 * (a + (n + 1) * d)\r\n           : by ring_nf\r\n     ... = ((n + 1) * (2 * a + n * d)) + 2 * (a + (n + 1) * d)\r\n           : by {congr; rw HI}\r\n     ... = (n + 2) * (2 * a + (n + 1) * d)\r\n           : by ring_nf\r\n     ... = (succ n + 1) * (2 * a + succ n * d)\r\n           : congr_arg2 (*) (by norm_cast) rfl, },\r\nend\r\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Suma_de_progresion_aritmetica.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;lean&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n<p>[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n<pre lang=\"isar\">\r\ntheory Suma_de_progresion_aritmetica\r\nimports Main HOL.Real\r\nbegin\r\n\r\nfun sumaPA :: \"real \u21d2 real \u21d2 nat \u21d2 real\" where\r\n  \"sumaPA a d 0 = a\"\r\n| \"sumaPA a d (Suc n) = sumaPA a d n + (a + (n + 1) * d)\"\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\nlemma\r\n  \"2 * sumaPA a d n = (n + 1) * (2 * a + n * d)\"\r\nproof (induct n)\r\n  show \"2 * sumaPA a d 0 =\r\n        (real 0 + 1) * (2 * a + real 0 * d)\"\r\n    by simp\r\nnext\r\n  fix n\r\n  assume HI : \"2 * sumaPA a d n =\r\n               (n + 1) * (2 * a + n * d)\"\r\n  have \"2 * sumaPA a d (Suc n) =\r\n        2 * (sumaPA a d n + (a + (n + 1) * d))\"\r\n    by simp\r\n  also have \"\u2026 = 2 * sumaPA a d n + 2 * (a + (n + 1) * d)\"\r\n    by simp\r\n  also have \"\u2026 = (n + 1) * (2 * a + n * d) + 2 * (a + (n + 1) * d)\"\r\n    using HI by simp\r\n  also have \"\u2026 = (real (Suc n) + 1) * (2 * a + (Suc n) * d)\"\r\n    by (simp add: algebra_simps)\r\n  finally show \"2 * sumaPA a d (Suc n) =\r\n                (real (Suc n) + 1) * (2 * a + (Suc n) * d)\"\r\n    by this\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\nlemma\r\n  \"2 * sumaPA a d n = (n + 1) * (2*a + n*d)\"\r\nproof (induct n)\r\n  case 0\r\n  then show ?case by simp\r\nnext\r\n  case (Suc n)\r\n  then show ?case by (simp add: algebra_simps)\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\nlemma\r\n  \"2 * sumaPA a d n = (n + 1) * (2*a + n*d)\"\r\nby (induct n) (simp_all add: algebra_simps)\r\n\r\nend\r\n<\/pre>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que la suma de los t\u00e9rminos de la progresi\u00f3n aritm\u00e9tica a + (a + d) + (a + 2 \u00d7 d) + \u00b7\u00b7\u00b7 + (a + n \u00d7 d) es (n + 1) \u00d7 (2 \u00d7 a + n \u00d7 d) \/ 2. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic open nat variable (n : \u2115) variables (a d : \u211d) @[simp] def sumaPA : \u211d \u2192 \u211d \u2192 \u2115 \u2192 \u211d | a d 0 := a | a d (n + 1) := sumaPA a d n + (a + (n + 1) * d) example : 2 * sumaPA a d n = (n + 1) * (2 * a + n * d) := sorry [expand title=\u00bbSoluciones&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[280],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/772"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/comments?post=772"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/772\/revisions"}],"predecessor-version":[{"id":773,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/772\/revisions\/773"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}