{"id":862,"date":"2025-12-05T10:11:12","date_gmt":"2025-12-05T09:11:12","guid":{"rendered":"https:\/\/mymatma.pl\/?p=862"},"modified":"2025-12-05T11:36:30","modified_gmt":"2025-12-05T10:36:30","slug":"zadanie-18","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=862","title":{"rendered":"Zadanie 18"},"content":{"rendered":"\n<p><strong>zadanie 30 \u2013 czerwiec 2019 (zad. Otwarte) (2 pkt)<\/strong><\/p>\n\n\n\n<p>W ci\u0105gu geometrycznym przez S<sub>n<\/sub> oznaczamy sum\u0119 n pocz\u0105tkowych wyraz\u00f3w tego ci\u0105gu, dla liczb naturalnych n\u22651 . Wiadomo, \u017ce dla pewnego ci\u0105gu geometrycznego: S<sub>1<\/sub>=2 i S<sub>2<\/sub>=12 . Wyznacz iloraz i pi\u0105ty wyraz tego ci\u0105gu. Przyznaj sobie samodzielnie punkty, zgodnie z proponowan\u0105 punktacj\u0105 ( 0 pkt,1 pkt, 2 pkt)<\/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=\"premium-wrapper-odp\" id=\"odp_69e226e2bb40a\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e226e2bb40a\"\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 iloraz = 5, a5 = 1250\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_69e226e2bb415\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e226e2bb415\"\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:760px;margin:16px auto;box-shadow:0 6px 20px rgba(18,38,63,0.06)\">\n\n  <p>W pewnym ci\u0105gu geometrycznym oznaczamy przez <strong>S<sub>n<\/sub><\/strong> sum\u0119 pierwszych <em>n<\/em> wyraz\u00f3w. Dane s\u0105 warto\u015bci:<\/p>\n\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    S\u2081 = 2 &nbsp;&nbsp;&nbsp; oraz &nbsp;&nbsp;&nbsp; S\u2082 = 12\n  <\/p>\n\n  <p><strong>Krok 1 \u2014 interpretacja danych:<\/strong><\/p>\n  <p>\u2022 S\u2081 = a\u2081 = 2<\/p>\n  <p>\u2022 S\u2082 = a\u2081 + a\u2082 = 12<\/p>\n\n  <p><strong>Krok 2 \u2014 wyznacz drugi wyraz a\u2082:<\/strong><\/p>\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    a\u2081 + a\u2082 = 12 <br>\n    2 + a\u2082 = 12  \u21d2  a\u2082 = 10\n  <\/p>\n\n  <p><strong>Krok 3 \u2014 oblicz iloraz r:<\/strong><\/p>\n  <p>a\u2082 = a\u2081 \u00b7 r, wi\u0119c:<\/p>\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    10 = 2r  \u21d2  r = 5\n  <\/p>\n\n  <p><strong>Krok 4 \u2014 znajd\u017a pi\u0105ty wyraz a\u2085:<\/strong><\/p>\n  <p>Wz\u00f3r og\u00f3lny ci\u0105gu geometrycznego: <br>\n     a\u2099 = a\u2081 \u00b7 r\u207d\u207f\u207b\u00b9\u207e\n  <\/p>\n\n  <p style=\"background:#fbfdff;padding:8px;border-radius:6px;border:1px solid #eef6ff;display:inline-block\">\n    a\u2085 = 2 \u00b7 5\u2074 = 2 \u00b7 625 = 1250\n  <\/p>\n\n  <div style=\"margin-top:14px;padding:12px;border-radius:8px;background:#f0faf0;border:1px solid #d8f1d8;color:#066a06;font-weight:600\">\n    Wynik: iloraz r = <strong>5<\/strong>, pi\u0105ty wyraz a\u2085 = <strong>1250<\/strong>.\n  <\/div>\n\n  <hr style=\"margin:22px 0;border:none;border-top:1px solid #eee\">\n\n  <h3 style=\"font-size:1.1rem;margin:10px 0 6px 0\">Samodzielna punktacja<\/h3>\n\n  <p>Zgodnie z proponowan\u0105 punktacj\u0105 (0 pkt, 1 pkt, 2 pkt) oce\u0144 swoje rozwi\u0105zanie:<\/p>\n\n  <ul style=\"line-height:1.6;margin-left:18px\">\n    <li><strong>2 pkt<\/strong> \u2014 masz poprawnie obliczony zar\u00f3wno iloraz r, jak i wyraz a\u2085.<\/li>\n    <li><strong>1 pkt<\/strong> \u2014 poprawnie masz tylko r <em>albo<\/em> tylko a\u2085.<\/li>\n    <li><strong>0 pkt<\/strong> \u2014 oba wyniki s\u0105 b\u0142\u0119dne lub zadanie nie zosta\u0142o rozwi\u0105zane.<\/li>\n  <\/ul>\n\n<\/div>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 30 \u2013 czerwiec 2019 (zad. Otwarte) (2 pkt) W ci\u0105gu geometrycznym przez Sn oznaczamy sum\u0119 n pocz\u0105tkowych wyraz\u00f3w tego ci\u0105gu, dla liczb naturalnych n\u22651 . Wiadomo, \u017ce dla pewnego ci\u0105gu geometrycznego: S1=2 i S2=12 . Wyznacz iloraz i pi\u0105ty wyraz tego ci\u0105gu. Przyznaj sobie samodzielnie punkty, zgodnie z proponowan\u0105 punktacj\u0105 ( 0 pkt,1 pkt, [&hellip;]<\/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":[31,66,49,81,83],"tags":[29,65,52],"class_list":["post-862","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-czerwiec-2019","category-iloraz-ciagu","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-czerwiec-2019","tag-iloraz-ciagu"],"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 30 \u2013 czerwiec 2019 (zad. Otwarte) (2 pkt) W ci\u0105gu geometrycznym przez Sn oznaczamy sum\u0119 n pocz\u0105tkowych wyraz\u00f3w tego ci\u0105gu, dla liczb naturalnych n\u22651 . Wiadomo, \u017ce dla pewnego ci\u0105gu geometrycznego: S1=2 i S2=12 . Wyznacz iloraz i pi\u0105ty wyraz tego ci\u0105gu. Przyznaj sobie samodzielnie punkty, zgodnie z proponowan\u0105 punktacj\u0105 ( 0 pkt,1 pkt,&hellip;","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/862","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=862"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/862\/revisions"}],"predecessor-version":[{"id":863,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/862\/revisions\/863"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=862"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=862"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}