{"id":905,"date":"2025-12-05T13:14:02","date_gmt":"2025-12-05T12:14:02","guid":{"rendered":"https:\/\/mymatma.pl\/?p=905"},"modified":"2025-12-05T14:22:14","modified_gmt":"2025-12-05T13:22:14","slug":"zadanie-31","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=905","title":{"rendered":"Zadanie 31"},"content":{"rendered":"\n<p><strong>zadanie 13 \u2013 maj 2018 (1 pkt)<\/strong><\/p>\n\n\n\n<p>Dany jest ci\u0105g geometryczny (a<sub>n<\/sub>) , okre\u015blony dla n\u22651 , w kt\u00f3rym a<sub>1<\/sub>=\u221a2 , a<sub>2<\/sub>=2\u221a2 , a<sub>3<\/sub>=4\u221a2 . Wz\u00f3r na n -ty wyraz tego ci\u0105gu ma posta\u0107:<\/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_69e22a9974c3d\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22a9974c3d\"\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_69e22a9974c4a\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22a9974c4a\"\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<!DOCTYPE html>\n<html lang=\"pl\">\n<head>\n    <meta charset=\"UTF-8\">\n    <meta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\">\n<\/head>\n<body>\n    <p>Dany jest ci\u0105g geometryczny (a<sub>n<\/sub>) o wyrazach:<\/p>\n    <ul>\n        <li>a<sub>1<\/sub> = \u221a2<\/li>\n        <li>a<sub>2<\/sub> = 2\u221a2<\/li>\n        <li>a<sub>3<\/sub> = 4\u221a2<\/li>\n    <\/ul>\n\n    <p>Wz\u00f3r na n-ty wyraz tego ci\u0105gu ma posta\u0107:<\/p>\n    <ul>\n        <li>a) a<sub>n<\/sub> = (\u221a2)<sup>n<\/sup><\/li>\n        <li>b) a<sub>n<\/sub> = (2n)\/(\u221a2)<\/li>\n        <li>c) a<sub>n<\/sub> = ((\u221a2)\/2)<sup>n<\/sup><\/li>\n        <li>d) a<sub>n<\/sub> = ((\u221a2)<sup>n<\/sup>)\/2<\/li>\n    <\/ul>\n\n    <div>\n        <p>Krok 1: Wyznaczamy iloraz ci\u0105gu:<\/p>\n        <p>q = a<sub>2<\/sub> \/ a<sub>1<\/sub> = (2\u221a2) \/ (\u221a2) = 2<\/p>\n\n        <p>Krok 2: Wz\u00f3r og\u00f3lny ci\u0105gu geometrycznego:<\/p>\n        <p>a<sub>n<\/sub> = a<sub>1<\/sub> \u22c5 q<sup>n-1<\/sup> = \u221a2 \u22c5 2<sup>n-1<\/sup><\/p>\n\n        <p>Przekszta\u0142camy wz\u00f3r do postaci pot\u0119gnej:<\/p>\n        <p>\u221a2 \u22c5 2<sup>n-1<\/sup> = \u221a2 \u22c5 (2<sup>n<\/sup> \/ 2) = (2<sup>n<\/sup> \u22c5 \u221a2) \/ 2 = ((\u221a2)<sup>n<\/sup>) \/ 2<\/p>\n        \n        <p>Poprawna forma odpowiada opcji:<\/p>\n        <p><strong>d) a<sub>n<\/sub> = ((\u221a2)<sup>n<\/sup>)\/2<\/strong><\/p>\n    <\/div>\n<\/body>\n<\/html>\n        <\/div>\n    <\/div>\n    \n\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 13 \u2013 maj 2018 (1 pkt) Dany jest ci\u0105g geometryczny (an) , okre\u015blony dla n\u22651 , w kt\u00f3rym a1=\u221a2 , a2=2\u221a2 , a3=4\u221a2 . Wz\u00f3r na n -ty wyraz tego ci\u0105gu ma posta\u0107:<\/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":[31,96,32,81,83],"tags":[29,95,93],"class_list":["post-905","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-maj-2018","category-obliczanie-wyrazow-ciagu-geometrycznego","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-maj-2018","tag-wyznaczanie-wzoru-ogolnego-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 13 \u2013 maj 2018 (1 pkt) Dany jest ci\u0105g geometryczny (an) , okre\u015blony dla n\u22651 , w kt\u00f3rym a1=\u221a2 , a2=2\u221a2 , a3=4\u221a2 . Wz\u00f3r na n -ty wyraz tego ci\u0105gu ma posta\u0107:","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/905","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=905"}],"version-history":[{"count":2,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/905\/revisions"}],"predecessor-version":[{"id":929,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/905\/revisions\/929"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=905"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=905"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=905"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}