{"id":2464,"date":"2022-04-25T15:42:23","date_gmt":"2022-04-25T15:42:23","guid":{"rendered":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/faktorial-reseny-priklad\/"},"modified":"2022-04-25T16:13:16","modified_gmt":"2022-04-25T16:13:16","slug":"faktorial-reseny-priklad","status":"publish","type":"lesson","link":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/faktorial-reseny-priklad\/","title":{"rendered":"Faktori\u00e1l \u0159e\u0161en\u00fd p\u0159\u00edklad"},"content":{"rendered":"<p><strong>Zad\u00e1n\u00ed:<\/strong> Zjednodu\u0161 n\u00e1sleduj\u00edc\u00ed v\u00fdraz:<\/p>\n<p>$$\\frac{(n+1)!}{n!}-\\frac{(2n)!}{(2n+1)!}+\\frac{(3n-1)!}{(3n-2)!}$$<\/p>\n<p><strong>\u0158e\u0161en\u00ed:<\/strong><\/p>\n<p style=\"text-align: center\">Proto\u017ee v\u00edme, \u017ee:<\/p>\n<p style=\"text-align: center\">$$(n+1)!=(n+1)\\cdot{n!}$$<\/p>\n<p style=\"text-align: center\">uprav\u00edme v\u00fdraz tak, abychom se zbavili faktori\u00e1l\u016f<\/p>\n<p>$$\\frac{(n+1)!}{n!}-\\frac{(2n)!}{(2n+1)!}+\\frac{(3n-1)!}{(3n-2)!}=$$<\/p>\n<p>$$=\\frac{(n+1){n!}}{n!}-\\frac{(2n)!}{(2n+1)(2n)!}+\\frac{(3n-1)(3n-2)!}{(3n-2)!}$$<\/p>\n<p style=\"text-align: center\">d\u00edky tomu jsme se zbavili faktori\u00e1l\u016f a dostali jsme:<\/p>\n<p>$$n+1-\\frac{1}{2n+1}+3n-1$$<\/p>\n<p style=\"text-align: center\">po \u00faprav\u011b:<\/p>\n<p>$$4n-\\frac{1}{2n+1}$$<\/p>\n<p style=\"text-align: center\">p\u0159evedeme na spole\u010dn\u00fd jmenovatel:<\/p>\n<p>$$\\frac{4n(2n+1)-1}{2n+1}$$<\/p>\n<p style=\"text-align: center\">po fin\u00e1ln\u00edm vyn\u00e1soben\u00ed z\u00e1vorky:<\/p>\n<p>$$\\frac{8n^{2}+4n-1}{2n+1}$$<\/p>\n<p style=\"text-align: center\">pro <strong><em>n<\/em><\/strong> mus\u00ed platit, \u017ee je z oboru p\u0159irozen\u00fdch \u010d\u00edsel<\/p>\n","protected":false},"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-2464","lesson","type-lesson","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Faktori\u00e1l \u0159e\u0161en\u00fd p\u0159\u00edklad | Studijni-svet.cz<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/faktorial-reseny-priklad\/\" \/>\n<meta property=\"og:locale\" content=\"cs_CZ\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Faktori\u00e1l \u0159e\u0161en\u00fd p\u0159\u00edklad | Studijni-svet.cz\" \/>\n<meta property=\"og:description\" content=\"Zad\u00e1n\u00ed: Zjednodu\u0161 n\u00e1sleduj\u00edc\u00ed v\u00fdraz: $$frac{(n+1)!}{n!}-frac{(2n)!}{(2n+1)!}+frac{(3n-1)!}{(3n-2)!}$$ \u0158e\u0161en\u00ed: Proto\u017ee v\u00edme, \u017ee: $$(n+1)!=(n+1)cdot{n!}$$ uprav\u00edme v\u00fdraz tak, abychom se zbavili faktori\u00e1l\u016f $$frac{(n+1)!}{n!}-frac{(2n)!}{(2n+1)!}+frac{(3n-1)!}{(3n-2)!}=$$ $$=frac{(n+1){n!}}{n!}-frac{(2n)!}{(2n+1)(2n)!}+frac{(3n-1)(3n-2)!}{(3n-2)!}$$ d\u00edky tomu jsme se zbavili faktori\u00e1l\u016f a dostali jsme: $$n+1-frac{1}{2n+1}+3n-1$$ po \u00faprav\u011b: $$4n-frac{1}{2n+1}$$ p\u0159evedeme na spole\u010dn\u00fd jmenovatel: $$frac{4n(2n+1)-1}{2n+1}$$ po fin\u00e1ln\u00edm vyn\u00e1soben\u00ed z\u00e1vorky: $$frac{8n^{2}+4n-1}{2n+1}$$ pro n mus\u00ed platit, \u017ee je z oboru p\u0159irozen\u00fdch \u010d\u00edsel\" \/>\n<meta property=\"og:url\" content=\"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/faktorial-reseny-priklad\/\" \/>\n<meta property=\"og:site_name\" content=\"Studijni-svet.cz\" \/>\n<meta property=\"article:publisher\" 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