{"id":858,"date":"2025-12-05T10:05:46","date_gmt":"2025-12-05T09:05:46","guid":{"rendered":"https:\/\/mymatma.pl\/?p=858"},"modified":"2025-12-05T11:36:40","modified_gmt":"2025-12-05T10:36:40","slug":"zadanie-17","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=858","title":{"rendered":"Zadanie 17"},"content":{"rendered":"\n<p><strong>zadanie 12 &#8211; maj 2019 (1 pkt)<\/strong><br>Dany jest ci\u0105g geometryczny a<sub>n<\/sub> , okre\u015blony dla n\u22651 . Wszystkie wyrazy tego ci\u0105gu s\u0105 dodatnie i spe\u0142niony jest warunek a<sub>5<\/sub>\/a<sub>3<\/sub>=1\/9 . 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_69e22ac03d2e5\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22ac03d2e5\"\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 A. \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_69e22ac03d2f2\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22ac03d2f2\"\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\" 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. Dane: <strong>a<sub>5<\/sub>\/a<sub>3<\/sub> = 1\/9<\/strong>. Obliczamy iloraz <em>r<\/em>.<\/p>\n\n  <p><strong>Krok 1 \u2014 zapisz wyra\u017cenia:<\/strong><\/p>\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    a<sub>5<\/sub> = a<sub>1<\/sub>\u00b7r\u2074, &nbsp; a<sub>3<\/sub> = a<sub>1<\/sub>\u00b7r\u00b2\n  <\/p>\n\n  <p><strong>Krok 2 \u2014 podziela<sub>5<\/sub> przez a<sub>3<\/sub> aby wyeliminowa\u0107 a<sub>1<\/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_3}=\\dfrac{a_1 r^4}{a_1 r^2}=r^2\\). Zatem \\(r^2=\\dfrac{1}{9}\\).\n  <\/p>\n\n  <p><strong>Krok 3 \u2014 oblicz r:<\/strong><\/p>\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    \\(r=\\pm\\dfrac{1}{3}\\). Poniewa\u017c wszystkie wyrazy ci\u0105gu s\u0105 dodatnie, wybieramy <em>r<\/em> dodatnie: <strong>r = 1\/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 #ddd;background:#fafafa\">a) 1\/3<\/span>\n    <span style=\"display:inline-block;margin-right:10px;padding:8px 10px;border-radius:6px;border:1px solid #ddd;background:#fafafa\">b) 1\/\u221a3<\/span>\n    <span style=\"display:inline-block;margin-right:10px;padding:8px 10px;border-radius:6px;border:1px solid #ddd;background:#fafafa\">c) 3<\/span>\n    <span style=\"display:inline-block;margin-right:10px;padding:8px 10px;border-radius:6px;border:1px solid #ddd;background:#fafafa\">d) \u221a3<\/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) 1\/3<\/strong>\n  <\/div>\n\n  <p style=\"color:#555;font-size:0.95rem;margin-top:8px\">Uwaga: dzielenie wyraz\u00f3w jest wygodnym sposobem na pozbycie si\u0119 nieznanego wyrazu pocz\u0105tkowego <code>a<sub>1<\/sub><\/code> i uzyskanie r\u00f3wnania tylko z <em>r<\/em>.<\/p>\n<\/div>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 12 &#8211; maj 2019 (1 pkt)Dany jest ci\u0105g geometryczny an , okre\u015blony dla n\u22651 . Wszystkie wyrazy tego ci\u0105gu s\u0105 dodatnie i spe\u0142niony jest warunek a5\/a3=1\/9 . 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":[31,49,64,81,83],"tags":[29,52,63],"class_list":["post-858","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-iloraz-ciagu","category-maj-2019","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-iloraz-ciagu","tag-maj-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; maj 2019 (1 pkt)Dany jest ci\u0105g geometryczny an , okre\u015blony dla n\u22651 . Wszystkie wyrazy tego ci\u0105gu s\u0105 dodatnie i spe\u0142niony jest warunek a5\/a3=1\/9 . Iloraz tego ci\u0105gu jest r\u00f3wny:","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/858","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=858"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/858\/revisions"}],"predecessor-version":[{"id":859,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/858\/revisions\/859"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=858"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=858"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=858"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}