{"id":855,"date":"2025-12-05T10:02:44","date_gmt":"2025-12-05T09:02:44","guid":{"rendered":"https:\/\/mymatma.pl\/?p=855"},"modified":"2025-12-05T11:36:51","modified_gmt":"2025-12-05T10:36:51","slug":"zadanie-16","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=855","title":{"rendered":"Zadanie 16"},"content":{"rendered":"\n<p><strong>zadanie 12 &#8211; sierpie\u0144 2019 (1 pkt)<\/strong><\/p>\n\n\n\n<p>Wszystkie wyrazy ci\u0105gu geometrycznego (a<sub>n<\/sub>) , okre\u015blonego dla n\u22651 , s\u0105 liczbami dodatnimi. Drugi wyraz tego ci\u0105gu jest r\u00f3wny 162 , a pi\u0105ty wyraz jest r\u00f3wny 48 . Oznacza to, \u017ce iloraz tego ci\u0105gu jest r\u00f3wny:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n    <div class=\"premium-wrapper-odp\" id=\"odp_69e22a2931611\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22a2931611\"\n                style=\"margin-top:10px; padding:12px 20px; background:#28a745; color:#fff; border-radius:25px; border:none; cursor:pointer; font-weight:bold;\">\n            Poka\u017c odpowied\u017a\n        <\/button>\n\n        <div class=\"premium-content-odp\" style=\"display:none; padding:15px; border:1px solid #28a745; margin-top:10px; background:#e6ffe6; border-radius:25px;\">\n            \nPrawid\u0142owa odpowied\u017a to 2\/3.\n        <\/div>\n    <\/div>\n    \n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n    <div class=\"premium-wrapper-wyjasnienie\" id=\"wyjasnienie_69e22a2931626\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22a2931626\"\n                style=\"margin-top:10px; padding:12px 20px; background:#28a745; color:#fff; border-radius:25px; border:none; cursor:pointer; font-weight:bold;\">\n            Poka\u017c wyja\u015bnienie\n        <\/button>\n\n        <div class=\"premium-content-wyjasnienie\" style=\"display:none; padding:15px; border:1px solid #28a745; margin-top:10px; background:#e6ffe6; border-radius:25px;\">\n            \n<div class=\"card\" role=\"article\" aria-labelledby=\"title\" style=\"font-family:system-ui,-apple-system,Segoe UI,Roboto,'Helvetica Neue',Arial;line-height:1.5;color:#111;padding:18px;background:#fff;border:1px solid #e2e6ea;border-radius:10px;max-width:720px;margin:16px auto;box-shadow:0 6px 20px rgba(18,38,63,0.06)\">\n  \n  <p>W ci\u0105gu geometrycznym <code>(a<sub>n<\/sub>)<\/code> wszystkie wyrazy s\u0105 dodatnie. Wiadomo, \u017ce <strong>a<sub>2<\/sub> = 162<\/strong> oraz <strong>a<sub>5<\/sub> = 48<\/strong>. Obliczamy iloraz <em>r<\/em> tego ci\u0105gu.<\/p>\n\n  <p><strong>Krok 1 \u2014 zapisz zale\u017cno\u015bci:<\/strong><\/p>\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    a<sub>2<\/sub> = a<sub>1<\/sub>\u00b7r = 162<br>\n    a<sub>5<\/sub> = a<sub>1<\/sub>\u00b7r<sup>4<\/sup> = 48\n  <\/p>\n\n  <p><strong>Krok 2 \u2014 dzielimy a<sub>5<\/sub> przez a<sub>2<\/sub>:<\/strong><\/p>\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    \\(\\dfrac{a_5}{a_2} = r^3\\) &nbsp;\u21d2&nbsp; \\(r^3 = \\dfrac{48}{162}\\)\n  <\/p>\n\n  <p><strong>Krok 3 \u2014 upraszczamy u\u0142amek:<\/strong><\/p>\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    \\(\\dfrac{48}{162} = \\dfrac{8}{27}\\) <br>\n    \\(r^3 = \\dfrac{8}{27}\\) \u21d2 \\(r = \\dfrac{2}{3}\\)\n  <\/p>\n\n  <p><strong>Krok 4 \u2014 wyb\u00f3r znaku:<\/strong> ci\u0105g ma tylko warto\u015bci dodatnie, wi\u0119c iloraz <em>r<\/em> r\u00f3wnie\u017c musi by\u0107 dodatni. Ostatecznie:<br>\n    <strong>r = 2\/3<\/strong>.\n  <\/p>\n\n  <div style=\"margin-top:14px\">\n    <span style=\"display:inline-block;margin-right:10px;padding:8px 10px;border-radius:6px;border:1px solid #1e90ff;background:#e9f4ff;font-weight:600\">a) 2\/3<\/span>\n    <span style=\"display:inline-block;margin-right:10px;padding:8px 10px;border-radius:6px;border:1px solid #ddd;background:#fafafa\">b) 3\/4<\/span>\n    <span style=\"display:inline-block;margin-right:10px;padding:8px 10px;border-radius:6px;border:1px solid #ddd;background:#fafafa\">c) 1\/3<\/span>\n    <span style=\"display:inline-block;margin-right:10px;padding:8px 10px;border-radius:6px;border:1px solid #ddd;background:#fafafa\">d) 1\/2<\/span>\n  <\/div>\n\n  <div style=\"margin-top:14px;padding:12px;border-radius:8px;background:#f0faf0;border:1px solid #d8f1d8;color:#066a06;font-weight:600\">\n    Poprawna odpowied\u017a: <strong>a) 2\/3<\/strong>\n  <\/div>\n<\/div>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 12 &#8211; sierpie\u0144 2019 (1 pkt) Wszystkie wyrazy ci\u0105gu geometrycznego (an) , okre\u015blonego dla n\u22651 , s\u0105 liczbami dodatnimi. Drugi wyraz tego ci\u0105gu jest r\u00f3wny 162 , a pi\u0105ty wyraz jest r\u00f3wny 48 . Oznacza to, \u017ce iloraz tego ci\u0105gu jest r\u00f3wny:<\/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":[31,49,62,81,83],"tags":[29,52,61],"class_list":["post-855","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-iloraz-ciagu","category-sierpien-2019","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-iloraz-ciagu","tag-sierpien-2019"],"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 12 &#8211; sierpie\u0144 2019 (1 pkt) Wszystkie wyrazy ci\u0105gu geometrycznego (an) , okre\u015blonego dla n\u22651 , s\u0105 liczbami dodatnimi. Drugi wyraz tego ci\u0105gu jest r\u00f3wny 162 , a pi\u0105ty wyraz jest r\u00f3wny 48 . Oznacza to, \u017ce iloraz tego ci\u0105gu jest r\u00f3wny:","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/855","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=855"}],"version-history":[{"count":2,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/855\/revisions"}],"predecessor-version":[{"id":861,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/855\/revisions\/861"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=855"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=855"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=855"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}