{"id":2639,"date":"2023-02-22T17:43:53","date_gmt":"2023-02-22T16:43:53","guid":{"rendered":"http:\/\/matma.com.pl\/?p=2639"},"modified":"2023-05-06T10:00:18","modified_gmt":"2023-05-06T08:00:18","slug":"zadanie-8-0-3-zbior-rozszerzenie","status":"publish","type":"post","link":"http:\/\/matma.com.pl\/?p=2639","title":{"rendered":"zadanie 8 (0-3) zbi\u00f3r rozszerzenie"},"content":{"rendered":"<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a=\\left&amp;space;(&amp;space;\\sqrt{5}+2&amp;space;\\right&amp;space;)^{2022}+\\left&amp;space;(&amp;space;\\sqrt{5}-2&amp;space;\\right&amp;space;)^{2022}\" alt=\"a=\\left ( \\sqrt{5}+2 \\right )^{2022}+\\left ( \\sqrt{5}-2 \\right )^{2022}\" align=\"absmiddle\" \/><\/p>\n<p>Pos\u0142u\u017cymy si\u0119 wzorem dwumianu Newtona:<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?{\\color{Orchid}&amp;space;\\left&amp;space;(&amp;space;a+b&amp;space;\\right&amp;space;)^n=\\binom{n}{0}a^nb^0+\\binom{n}{1}a^{n-1}b^1+\\binom{n}{2}a^{n-2}b^2+...+}\" alt=\"{\\color{Orchid} \\left ( a+b \\right )^n=\\binom{n}{0}a^nb^0+\\binom{n}{1}a^{n-1}b^1+\\binom{n}{2}a^{n-2}b^2+...+}\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?{\\color{Orchid}&amp;space;+...+\\binom{n}{n-2}a^2b^{n-2}+\\binom{n}{n-1}a^1b^{n-1}\\binom{n}{x}a^0b^n}\" alt=\"{\\color{Orchid} +...+\\binom{n}{n-2}a^2b^{n-2}+\\binom{n}{n-1}a^1b^{n-1}\\binom{n}{x}a^0b^n}\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;(&amp;space;\\sqrt{5}&amp;space;+2\\right&amp;space;)^{2022}=\\binom{2022}{0}\\sqrt{5}^{2022}2^0+\\binom{2022}{1}\\sqrt{5}^{2021}2^1+\\binom{2022}{2}\\sqrt{5}^{2020}2^2+\" alt=\"\\left ( \\sqrt{5} +2\\right )^{2022}=\\binom{2022}{0}\\sqrt{5}^{2022}2^0+\\binom{2022}{1}\\sqrt{5}^{2021}2^1+\\binom{2022}{2}\\sqrt{5}^{2020}2^2+\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?+...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}+\\binom{2022}{2021}\\sqrt{5}\\cdot&amp;space;2^{2021}+\\binom{2022}{2022}\\sqrt{5}^0\\cdot&amp;space;2^{2022}\" alt=\"+...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}+\\binom{2022}{2021}\\sqrt{5}\\cdot 2^{2021}+\\binom{2022}{2022}\\sqrt{5}^0\\cdot 2^{2022}\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;(&amp;space;\\sqrt{5}&amp;space;-2\\right&amp;space;)^{2022}=\\left&amp;space;(&amp;space;\\sqrt{5}&amp;space;+\\left&amp;space;(-2&amp;space;\\right&amp;space;)\\right&amp;space;)^{2022}\" alt=\"\\left ( \\sqrt{5} -2\\right )^{2022}=\\left ( \\sqrt{5} +\\left (-2 \\right )\\right )^{2022}\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;(&amp;space;\\sqrt{5}+&amp;space;\\left&amp;space;(-2&amp;space;\\right&amp;space;)\\right&amp;space;)^{2022}=\\binom{2022}{0}\\sqrt{5}^{2022}\\left&amp;space;(-2&amp;space;\\right&amp;space;)^0+\\binom{2022}{1}\\sqrt{5}^{2021}\\left&amp;space;(-2&amp;space;\\right&amp;space;)^1+\\binom{2022}{2}\\sqrt{5}^{2020}\\left&amp;space;(-2&amp;space;\\right&amp;space;)^2+\" alt=\"\\left ( \\sqrt{5}+ \\left (-2 \\right )\\right )^{2022}=\\binom{2022}{0}\\sqrt{5}^{2022}\\left (-2 \\right )^0+\\binom{2022}{1}\\sqrt{5}^{2021}\\left (-2 \\right )^1+\\binom{2022}{2}\\sqrt{5}^{2020}\\left (-2 \\right )^2+\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?+...+\\binom{2022}{2020}\\sqrt{5}^2\\left&amp;space;(&amp;space;-2&amp;space;\\right&amp;space;)^{2020}+\\binom{2022}{2021}\\sqrt{5}\\cdot&amp;space;\\left&amp;space;(&amp;space;-2&amp;space;\\right&amp;space;)^{2021}+\\binom{2022}{2022}\\sqrt{5}^0\\cdot&amp;space;\\left&amp;space;(&amp;space;-2&amp;space;\\right&amp;space;)^{2022}=\" alt=\"+...+\\binom{2022}{2020}\\sqrt{5}^2\\left ( -2 \\right )^{2020}+\\binom{2022}{2021}\\sqrt{5}\\cdot \\left ( -2 \\right )^{2021}+\\binom{2022}{2022}\\sqrt{5}^0\\cdot \\left ( -2 \\right )^{2022}=\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?=\\binom{2022}{0}\\sqrt{5}^{2022}2^0-\\binom{2022}{1}\\sqrt{5}^{2021}2^1+\\binom{2022}{2}\\sqrt{5}^{2020}2^2-\" alt=\"=\\binom{2022}{0}\\sqrt{5}^{2022}2^0-\\binom{2022}{1}\\sqrt{5}^{2021}2^1+\\binom{2022}{2}\\sqrt{5}^{2020}2^2-\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}-\\binom{2022}{2021}\\sqrt{5}\\cdot&amp;space;2^{2021}+\\binom{2022}{2022}\\sqrt{5}^0\\cdot&amp;space;2^{2022}\" alt=\"-...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}-\\binom{2022}{2021}\\sqrt{5}\\cdot 2^{2021}+\\binom{2022}{2022}\\sqrt{5}^0\\cdot 2^{2022}\" align=\"absmiddle\" \/><\/p>\n<p>Wr\u00f3\u0107my do <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a\" alt=\"a\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a=\\left&amp;space;(&amp;space;\\sqrt{5}+2&amp;space;\\right&amp;space;)^{2022}+\\left&amp;space;(&amp;space;\\sqrt{5}-2&amp;space;\\right&amp;space;)^{2022}=\\binom{2022}{0}\\sqrt{5}^{2022}2^0+\\cancel{\\binom{2022}{1}\\sqrt{5}^{2021}2^1}+\" alt=\"a=\\left ( \\sqrt{5}+2 \\right )^{2022}+\\left ( \\sqrt{5}-2 \\right )^{2022}=\\binom{2022}{0}\\sqrt{5}^{2022}2^0+\\cancel{\\binom{2022}{1}\\sqrt{5}^{2021}2^1}+\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?+\\binom{2022}{2}\\sqrt{5}^{2020}2^2+...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}+\\cancel{\\binom{2022}{2021}\\sqrt{5}\\cdot&amp;space;2^{2021}}+\" alt=\"+\\binom{2022}{2}\\sqrt{5}^{2020}2^2+...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}+\\cancel{\\binom{2022}{2021}\\sqrt{5}\\cdot 2^{2021}}+\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?+\\binom{2022}{2022}\\sqrt{5}^0\\cdot&amp;space;2^{2022}+\\binom{2022}{0}\\sqrt{5}^{2022}2^0-\\cancel{\\binom{2022}{1}\\sqrt{5}^{2021}2^1}+\\binom{2022}{2}\\sqrt{5}^{2020}2^2-\" alt=\"+\\binom{2022}{2022}\\sqrt{5}^0\\cdot 2^{2022}+\\binom{2022}{0}\\sqrt{5}^{2022}2^0-\\cancel{\\binom{2022}{1}\\sqrt{5}^{2021}2^1}+\\binom{2022}{2}\\sqrt{5}^{2020}2^2-\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}-\\cancel{\\binom{2022}{2021}\\sqrt{5}\\cdot&amp;space;2^{2021}}+\\binom{2022}{2022}\\sqrt{5}^0\\cdot&amp;space;2^{2022}\" alt=\"-...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}-\\cancel{\\binom{2022}{2021}\\sqrt{5}\\cdot 2^{2021}}+\\binom{2022}{2022}\\sqrt{5}^0\\cdot 2^{2022}\" align=\"absmiddle\" \/><\/p>\n<p>Widzimy, \u017ce ka\u017cde wyrazy z nieparzystymi pot\u0119gami (zar\u00f3wno przy <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\sqrt{5}\" alt=\"\\sqrt{5}\" align=\"absmiddle\" \/> jak i przy <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?2\" alt=\"2\" align=\"absmiddle\" \/>) si\u0119 zredukuj\u0105. Pozostanie nam:<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a=\\binom{2022}{0}\\sqrt{5}^{2022}2^0+\\binom{2022}{2}\\sqrt{5}^{2020}2^2+...+\\binom{2022}{2020}\\sqrt{5}^{2}2^{2020}+\\binom{2022}{2022}\\sqrt{5}^02^{2022}+\" alt=\"a=\\binom{2022}{0}\\sqrt{5}^{2022}2^0+\\binom{2022}{2}\\sqrt{5}^{2020}2^2+...+\\binom{2022}{2020}\\sqrt{5}^{2}2^{2020}+\\binom{2022}{2022}\\sqrt{5}^02^{2022}+\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?+\\binom{2022}{0}\\sqrt{5}^{2022}+\\binom{2022}{2}\\sqrt{5}^{2020}2^2+...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}+\\binom{2022}{2022}\\sqrt{5}^02^{2022}\" alt=\"+\\binom{2022}{0}\\sqrt{5}^{2022}+\\binom{2022}{2}\\sqrt{5}^{2020}2^2+...+\\binom{2022}{2020}\\sqrt{5}^22^{2020}+\\binom{2022}{2022}\\sqrt{5}^02^{2022}\" align=\"absmiddle\" \/><\/p>\n<p>Mo\u017cemy zauwa\u017cy\u0107, \u017ce pozosta\u0142y nam dwa takie same wyrazy przy <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\binom{2022}{0}\" alt=\"\\binom{2022}{0}\" align=\"absmiddle\" \/> oraz dwa takie same wyrazy przy <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\binom{2022}{2},...\\;&amp;space;,\\binom{2022}{2020}\" alt=\"\\binom{2022}{2},...\\; ,\\binom{2022}{2020}\" align=\"absmiddle\" \/> oraz <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\binom{2022}{2022}\" alt=\"\\binom{2022}{2022}\" align=\"absmiddle\" \/>. Uporz\u0105dkujmy nasze wyra\u017cenie<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?a=2\\cdot&amp;space;\\binom{2022}{0}\\sqrt{5}^{2022}2^0+2\\cdot&amp;space;\\binom{2022}{2}\\sqrt{5}^{2020}2^2+...+2\\cdot&amp;space;\\binom{2022}{2020}\\sqrt{5}^{2}2^{2020}+\" alt=\"a=2\\cdot \\binom{2022}{0}\\sqrt{5}^{2022}2^0+2\\cdot \\binom{2022}{2}\\sqrt{5}^{2020}2^2+...+2\\cdot \\binom{2022}{2020}\\sqrt{5}^{2}2^{2020}+\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?+2\\cdot&amp;space;\\binom{2022}{2022}\\sqrt{5}^02^{2022}=2\\cdot&amp;space;\\binom{2022}{0}5^{1011}\\cdot&amp;space;1+2\\cdot&amp;space;\\binom{2022}{2}5^{1010}\\cdot&amp;space;2+...+\" alt=\"+2\\cdot \\binom{2022}{2022}\\sqrt{5}^02^{2022}=2\\cdot \\binom{2022}{0}5^{1011}\\cdot 1+2\\cdot \\binom{2022}{2}5^{1010}\\cdot 2+...+\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?+2\\cdot&amp;space;\\binom{2022}{2020}5\\cdot&amp;space;2^{2022}+2\\cdot&amp;space;\\binom{2022}{2022}1\\cdot&amp;space;2^{2022}\" alt=\"+2\\cdot \\binom{2022}{2020}5\\cdot 2^{2022}+2\\cdot \\binom{2022}{2022}1\\cdot 2^{2022}\" align=\"absmiddle\" \/><\/p>\n<p>Widzimy teraz, \u017ce ka\u017cdy nasz wyraz jest liczba wymiern\u0105 parzyst\u0105. Zatem ca\u0142a suma jest liczb\u0105 wymiern\u0105 parzyst\u0105.<\/p>\n<p style=\"text-align: right;\">Co nale\u017ca\u0142o pokaza\u0107.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Pos\u0142u\u017cymy si\u0119 wzorem dwumianu Newtona: Wr\u00f3\u0107my do Widzimy, \u017ce ka\u017cde wyrazy z nieparzystymi pot\u0119gami (zar\u00f3wno przy jak i przy ) si\u0119 zredukuj\u0105. Pozostanie nam: Mo\u017cemy zauwa\u017cy\u0107, \u017ce pozosta\u0142y nam dwa takie same wyrazy przy oraz dwa takie same wyrazy przy oraz . Uporz\u0105dkujmy nasze wyra\u017cenie Widzimy teraz, \u017ce ka\u017cdy nasz wyraz jest liczba wymiern\u0105 parzyst\u0105. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2639","post","type-post","status-publish","format-standard","hentry","category-najnowsze-pliki"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>zadanie 8 (0-3) zbi\u00f3r rozszerzenie - Matma.com.pl<\/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:\/\/matma.com.pl\/?p=2639\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"zadanie 8 (0-3) zbi\u00f3r rozszerzenie - Matma.com.pl\" \/>\n<meta property=\"og:description\" content=\"Pos\u0142u\u017cymy si\u0119 wzorem dwumianu Newtona: Wr\u00f3\u0107my do Widzimy, \u017ce ka\u017cde wyrazy z nieparzystymi pot\u0119gami (zar\u00f3wno przy jak i przy ) si\u0119 zredukuj\u0105. Pozostanie nam: Mo\u017cemy zauwa\u017cy\u0107, \u017ce pozosta\u0142y nam dwa takie same wyrazy przy oraz dwa takie same wyrazy przy oraz . Uporz\u0105dkujmy nasze wyra\u017cenie Widzimy teraz, \u017ce ka\u017cdy nasz wyraz jest liczba wymiern\u0105 parzyst\u0105. 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Pozostanie nam: Mo\u017cemy zauwa\u017cy\u0107, \u017ce pozosta\u0142y nam dwa takie same wyrazy przy oraz dwa takie same wyrazy przy oraz . Uporz\u0105dkujmy nasze wyra\u017cenie Widzimy teraz, \u017ce ka\u017cdy nasz wyraz jest liczba wymiern\u0105 parzyst\u0105. 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