Ci\u0105g geometryczny<\/title>\n<\/head>\n<body>\n\nMamy ci\u0105g geometryczny <code>a\u2099<\/code> dla <code>n\u22651<\/code>, w kt\u00f3rym wszystkie wyrazy s\u0105 dodatnie. Wiemy, \u017ce:<\/p>\n\n<code>6a\u2081 \u2013 5a\u2082 + a\u2083 = 0<\/code><\/p>\n\nChcemy znale\u017a\u0107 iloraz <code>q<\/code> ci\u0105gu w przedziale <code>[2\u221a2, 3\u221a2]<\/code>.<\/p>\n\nKrok 1:<\/strong> Wyra\u017camy wyrazy przez iloraz:<\/p>\n<ul>\n<li><code>a\u2082 = a\u2081\u00b7q<\/code><\/li>\n<li><code>a\u2083 = a\u2081\u00b7q\u00b2<\/code><\/li>\n<\/ul>\n\nPodstawiamy do r\u00f3wnania:<\/p>\n<code>6a\u2081 \u2013 5(a\u2081\u00b7q) + (a\u2081\u00b7q\u00b2) = 0<\/code><\/p>\n\nWyci\u0105gamy <code>a\u2081<\/code> przed nawias:<\/p>\n<code>a\u2081 (6 \u2013 5q + q\u00b2) = 0<\/code><\/p>\n\nSkoro <code>a\u2081 \u2260 0<\/code>, mamy r\u00f3wnanie kwadratowe:<\/p>\n<code>q\u00b2 \u2013 5q + 6 = 0<\/code><\/p>\n\nKrok 2:<\/strong> Rozwi\u0105zujemy r\u00f3wnanie kwadratowe:<\/p>\n<code>(q \u2013 2)(q \u2013 3) = 0<\/code><\/p>\nSt\u0105d <code>q = 2<\/code> lub <code>q = 3<\/code>.<\/p>\n\nKrok 3:<\/strong> Sprawdzamy przedzia\u0142 <code>[2\u221a2, 3\u221a2]<\/code>:<\/p>\n<ul>\n<li><code>2\u221a2 \u2248 2,828<\/code><\/li>\n<li><code>3\u221a2 \u2248 4,243<\/code><\/li>\n<\/ul>\n\u017baden z wyraz\u00f3w <code>q = 2<\/code> ani <code>q = 3<\/code> nie mie\u015bci si\u0119 poza przedzia\u0142em? \n<code>2 < 2,828<\/code> \u2013 nie mie\u015bci si\u0119, <code>3<\/code> \u2013 mie\u015bci si\u0119, bo <code>2,828 \u2264 3 \u2264 4,243<\/code>.<\/p>\n\nOdpowied\u017a:<\/strong> <code>q = 3<\/code><\/p>\n\n<\/body>\n<\/html>\n\n <\/div>\n <\/div>\n \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"zadanie 33 \u2013 maj 2020 (zad. Otwarte) (4 pkt)Wszystkie wyrazy ci\u0105gu geometrycznego (an), okre\u015blonego dla n\u22651, s\u0105 dodatnie. Wyrazy tego ci\u0105gu spe\u0142niaj\u0105 warunek 6a1 \u2013 5a2 + a3 = 0. Oblicz iloraz q tego ci\u0105gu nale\u017c\u0105cy do przedzia\u0142u [2\u221a2, 3\u221a2].<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"saved_in_kubio":false,"_uag_custom_page_level_css":"","_themeisle_gutenberg_block_has_review":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31,49,84,81,83],"tags":[],"class_list":["post-745","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-iloraz-ciagu","category-maj-2020","category-zadania-maturalne","category-zadania-tematami"],"jetpack_featured_media_url":"","uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"kubio-fullhd":false,"woocommerce_thumbnail":false,"woocommerce_single":false,"woocommerce_gallery_thumbnail":false},"uagb_author_info":{"display_name":"adminmymatma","author_link":"https:\/\/mymatma.pl\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"zadanie 33 \u2013 maj 2020 (zad. Otwarte) (4 pkt)Wszystkie wyrazy ci\u0105gu geometrycznego (an), okre\u015blonego dla n\u22651, s\u0105 dodatnie. Wyrazy tego ci\u0105gu spe\u0142niaj\u0105 warunek 6a1 \u2013 5a2 + a3 = 0. Oblicz iloraz q tego ci\u0105gu nale\u017c\u0105cy do przedzia\u0142u [2\u221a2, 3\u221a2].","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/745","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=745"}],"version-history":[{"count":6,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/745\/revisions"}],"predecessor-version":[{"id":753,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/745\/revisions\/753"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=745"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=745"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=745"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

\n \n\n\n\n\nCi\u0105g geometryczny<\/title>\n<\/head>\n<body>\n\nMamy ci\u0105g geometryczny <code>a\u2099<\/code> dla <code>n\u22651<\/code>, w kt\u00f3rym wszystkie wyrazy s\u0105 dodatnie. Wiemy, \u017ce:<\/p>\n\n<code>6a\u2081 \u2013 5a\u2082 + a\u2083 = 0<\/code><\/p>\n\nChcemy znale\u017a\u0107 iloraz <code>q<\/code> ci\u0105gu w przedziale <code>[2\u221a2, 3\u221a2]<\/code>.<\/p>\n\nKrok 1:<\/strong> Wyra\u017camy wyrazy przez iloraz:<\/p>\n<ul>\n<li><code>a\u2082 = a\u2081\u00b7q<\/code><\/li>\n<li><code>a\u2083 = a\u2081\u00b7q\u00b2<\/code><\/li>\n<\/ul>\n\nPodstawiamy do r\u00f3wnania:<\/p>\n<code>6a\u2081 \u2013 5(a\u2081\u00b7q) + (a\u2081\u00b7q\u00b2) = 0<\/code><\/p>\n\nWyci\u0105gamy <code>a\u2081<\/code> przed nawias:<\/p>\n<code>a\u2081 (6 \u2013 5q + q\u00b2) = 0<\/code><\/p>\n\nSkoro <code>a\u2081 \u2260 0<\/code>, mamy r\u00f3wnanie kwadratowe:<\/p>\n<code>q\u00b2 \u2013 5q + 6 = 0<\/code><\/p>\n\nKrok 2:<\/strong> Rozwi\u0105zujemy r\u00f3wnanie kwadratowe:<\/p>\n<code>(q \u2013 2)(q \u2013 3) = 0<\/code><\/p>\nSt\u0105d <code>q = 2<\/code> lub <code>q = 3<\/code>.<\/p>\n\nKrok 3:<\/strong> Sprawdzamy przedzia\u0142 <code>[2\u221a2, 3\u221a2]<\/code>:<\/p>\n<ul>\n<li><code>2\u221a2 \u2248 2,828<\/code><\/li>\n<li><code>3\u221a2 \u2248 4,243<\/code><\/li>\n<\/ul>\n\u017baden z wyraz\u00f3w <code>q = 2<\/code> ani <code>q = 3<\/code> nie mie\u015bci si\u0119 poza przedzia\u0142em? \n<code>2 < 2,828<\/code> \u2013 nie mie\u015bci si\u0119, <code>3<\/code> \u2013 mie\u015bci si\u0119, bo <code>2,828 \u2264 3 \u2264 4,243<\/code>.<\/p>\n\nOdpowied\u017a:<\/strong> <code>q = 3<\/code><\/p>\n\n<\/body>\n<\/html>\n\n <\/div>\n <\/div>\n \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"zadanie 33 \u2013 maj 2020 (zad. Otwarte) (4 pkt)Wszystkie wyrazy ci\u0105gu geometrycznego (an), okre\u015blonego dla n\u22651, s\u0105 dodatnie. Wyrazy tego ci\u0105gu spe\u0142niaj\u0105 warunek 6a1 \u2013 5a2 + a3 = 0. Oblicz iloraz q tego ci\u0105gu nale\u017c\u0105cy do przedzia\u0142u [2\u221a2, 3\u221a2].<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"saved_in_kubio":false,"_uag_custom_page_level_css":"","_themeisle_gutenberg_block_has_review":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31,49,84,81,83],"tags":[],"class_list":["post-745","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-iloraz-ciagu","category-maj-2020","category-zadania-maturalne","category-zadania-tematami"],"jetpack_featured_media_url":"","uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"kubio-fullhd":false,"woocommerce_thumbnail":false,"woocommerce_single":false,"woocommerce_gallery_thumbnail":false},"uagb_author_info":{"display_name":"adminmymatma","author_link":"https:\/\/mymatma.pl\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"zadanie 33 \u2013 maj 2020 (zad. Otwarte) (4 pkt)Wszystkie wyrazy ci\u0105gu geometrycznego (an), okre\u015blonego dla n\u22651, s\u0105 dodatnie. Wyrazy tego ci\u0105gu spe\u0142niaj\u0105 warunek 6a1 \u2013 5a2 + a3 = 0. Oblicz iloraz q tego ci\u0105gu nale\u017c\u0105cy do przedzia\u0142u [2\u221a2, 3\u221a2].","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/745","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=745"}],"version-history":[{"count":6,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/745\/revisions"}],"predecessor-version":[{"id":753,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/745\/revisions\/753"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=745"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=745"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=745"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}