{"id":705,"date":"2025-12-04T08:17:32","date_gmt":"2025-12-04T07:17:32","guid":{"rendered":"http:\/\/mymatma.pl\/?p=705"},"modified":"2025-12-05T11:39:00","modified_gmt":"2025-12-05T10:39:00","slug":"zadanie-4","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=705","title":{"rendered":"Zadanie 4"},"content":{"rendered":"\n<p><br><strong>zadanie 14 czerwiec 2021 (1pkt)<\/strong><\/p>\n\n\n\n<p><br>Ci\u0105g geometryczny (a<sub>n<\/sub>) , okre\u015blony dla ka\u017cdej liczby naturalnej n\u22651 , jest rosn\u0105cy i wszystkie jego wyrazy s\u0105 dodatnie. Ponadto spe\u0142niony jest warunek a<sub>3<\/sub>=a<sub>1<\/sub>\u22c5a<sub>2<\/sub> . Niech q oznacza iloraz ci\u0105gu (a<sub>n<\/sub>) . Wtedy:<\/p>\n\n\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_69e22a56343b1\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22a56343b1\"\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 B.\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_69e22a56343be\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22a56343be\"\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>Dla ci\u0105gu geometrycznego mamy:<\/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\n  <p>Z warunku <code>a\u2083 = a\u2081\u00b7a\u2082<\/code> otrzymujemy:<\/p>\n  <p><code>a\u2081\u00b7q\u00b2 = a\u2081 \u00b7 (a\u2081\u00b7q)<\/code><\/p>\n  <p>Zak\u0142adaj\u0105c <code>a\u2081 \u2260 0<\/code> (wyrazy s\u0105 dodatnie), dzielimy przez <code>a\u2081<\/code> i otrzymujemy:<\/p>\n  <p><code>q\u00b2 = a\u2081\u00b7q<\/code><\/p>\n  <p>St\u0105d <code>q\u00b2 - a\u2081\u00b7q = 0<\/code>, czyli <code>q(q - a\u2081) = 0<\/code>.<\/p>\n  <p>Skoro <code>q \u2260 0<\/code>, musi by\u0107 <code>q - a\u2081 = 0<\/code>, czyli <strong><code>a\u2081 = q<\/code><\/strong>.<\/p>\n\n  <p>Uwaga: poniewa\u017c ci\u0105g jest rosn\u0105cy i dodatni, <code>q &gt; 1<\/code>, co jest zgodne z otrzymym wynikiem.<\/p>\n<\/div>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 14 czerwiec 2021 (1pkt) Ci\u0105g geometryczny (an) , okre\u015blony dla ka\u017cdej liczby naturalnej n\u22651 , jest rosn\u0105cy i wszystkie jego wyrazy s\u0105 dodatnie. Ponadto spe\u0142niony jest warunek a3=a1\u22c5a2 . Niech q oznacza iloraz ci\u0105gu (an) . Wtedy:<\/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":[31,36,32,81,83],"tags":[29,35,28],"class_list":["post-705","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-czerwiec-2021","category-obliczanie-wyrazow-ciagu-geometrycznego","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-czerwiec-2021","tag-obliczanie-wyrazow-ciagu-geometrycznego"],"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 14 czerwiec 2021 (1pkt) Ci\u0105g geometryczny (an) , okre\u015blony dla ka\u017cdej liczby naturalnej n\u22651 , jest rosn\u0105cy i wszystkie jego wyrazy s\u0105 dodatnie. Ponadto spe\u0142niony jest warunek a3=a1\u22c5a2 . Niech q oznacza iloraz ci\u0105gu (an) . Wtedy:","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/705","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=705"}],"version-history":[{"count":4,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/705\/revisions"}],"predecessor-version":[{"id":709,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/705\/revisions\/709"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=705"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=705"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=705"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}