{"id":741,"date":"2025-12-04T10:09:55","date_gmt":"2025-12-04T09:09:55","guid":{"rendered":"http:\/\/mymatma.pl\/?p=741"},"modified":"2025-12-05T11:37:09","modified_gmt":"2025-12-05T10:37:09","slug":"zadanie-14","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=741","title":{"rendered":"Zadanie 14"},"content":{"rendered":"\n<p><strong>zadanie 15 \u2013 maj 2022 (1 pkt)<\/strong><\/p>\n\n\n\n<p>Wszystkie wyrazy niesko\u0144czonego ci\u0105gu geometrycznego (a<sub>n<\/sub>) , okre\u015blonego dla ka\u017cdej liczby naturalnej n\u22651 , s\u0105 dodatnie i 9a<sub>5<\/sub>=4a<sub>3<\/sub> . Wtedy 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_69e22a788360a\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22a788360a\"\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_69e22a788361c\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22a788361c\"\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>\n  <p>Mamy niesko\u0144czony ci\u0105g geometryczny <code>(a\u2099)<\/code> z wyrazami dodatnimi i warunkiem <code>9\u00b7a\u2085 = 4\u00b7a\u2083<\/code>. Niech <code>q<\/code> b\u0119dzie ilorazem ci\u0105gu.<\/p>\n\n  <p>Dla ci\u0105gu geometrycznego <code>a\u2099 = a\u2081\u00b7q^(n\u22121)<\/code>, mamy:<\/p>\n  <ul>\n    <li><code>a\u2083 = a\u2081\u00b7q\u00b2<\/code><\/li>\n    <li><code>a\u2085 = a\u2081\u00b7q\u2074<\/code><\/li>\n  <\/ul>\n\n  <p>Podstawiamy warunek:<\/p>\n  <p><code>9\u00b7a\u2085 = 4\u00b7a\u2083 \u2192 9\u00b7(a\u2081\u00b7q\u2074) = 4\u00b7(a\u2081\u00b7q\u00b2)<\/code><\/p>\n\n  <p>Dzielimy przez <code>a\u2081 \u2260 0<\/code>:<\/p>\n  <p><code>9\u00b7q\u2074 = 4\u00b7q\u00b2 \u2192 9\u00b7q\u2074 \u2212 4\u00b7q\u00b2 = 0 \u2192 q\u00b2(9\u00b7q\u00b2 \u2212 4) = 0<\/code><\/p>\n\n  <p>Skoro wyrazy s\u0105 dodatnie, <code>q \u2260 0<\/code>, wi\u0119c:<\/p>\n  <p><code>9\u00b7q\u00b2 \u2212 4 = 0 \u2192 q\u00b2 = 4\/9 \u2192 q = 2\/3<\/code><\/p>\n\n  <p><strong>Poprawna odpowied\u017a: a) 2\/3<\/strong><\/p>\n<\/div>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 15 \u2013 maj 2022 (1 pkt) Wszystkie wyrazy niesko\u0144czonego ci\u0105gu geometrycznego (an) , okre\u015blonego dla ka\u017cdej liczby naturalnej n\u22651 , s\u0105 dodatnie i 9a5=4a3 . Wtedy 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":[31,49,58,81,83],"tags":[29,52,57],"class_list":["post-741","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-iloraz-ciagu","category-maj-2022","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-iloraz-ciagu","tag-maj-2022"],"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 15 \u2013 maj 2022 (1 pkt) Wszystkie wyrazy niesko\u0144czonego ci\u0105gu geometrycznego (an) , okre\u015blonego dla ka\u017cdej liczby naturalnej n\u22651 , s\u0105 dodatnie i 9a5=4a3 . Wtedy iloraz tego ci\u0105gu jest r\u00f3wny:","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/741","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=741"}],"version-history":[{"count":2,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/741\/revisions"}],"predecessor-version":[{"id":743,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/741\/revisions\/743"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=741"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=741"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=741"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}