{"id":1348,"date":"2025-12-23T14:50:06","date_gmt":"2025-12-23T13:50:06","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1348"},"modified":"2025-12-23T14:50:10","modified_gmt":"2025-12-23T13:50:10","slug":"zadanie-139","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1348","title":{"rendered":"Zadanie 139"},"content":{"rendered":"\n<p><strong>zadanie 16 \u2013 16.3 \u2013 wrzesie\u0144 2022 (3pkt)<\/strong><\/p>\n\n\n\n<p>Dany jest ci\u0105g (a<sub>n<\/sub>) okre\u015blony wzorem a<sub>n<\/sub>=3n\u22121 dla ka\u017cdej liczby naturalnej n\u22651. <\/p>\n\n\n\n<p><strong>zadanie 16.1.<\/strong> Doko\u0144cz zdanie tak, aby by\u0142o prawdziwe. Wybierz odpowied\u017a <strong>a, b<\/strong> albo <strong>c <\/strong>oraz jej uzasadnienie <strong>1., 2<\/strong>. albo <strong>3.<\/strong> Ci\u0105g a<sub>n<\/sub> jest:<\/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<p>a) rosn\u0105cy<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) malej\u0105cy<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) sta\u0142y<\/p>\n<\/div>\n<\/div>\n\n\n\n<p>Poniewa\u017c dla ka\u017cdej liczny naturalnej n\u22651:<\/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<p>1) a<sub>n+1<\/sub> &#8211; a<sub>n<\/sub>= -1<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>2) a<sub>n+1<\/sub> &#8211; a<sub>n<\/sub>= 0<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>3) a<sub>n+1<\/sub> &#8211; a<sub>n<\/sub>= 3<\/p>\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_69e22b4552e40\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b4552e40\"\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 <p>\na) rosn\u0105cy<br>\n3) a<sub>n+1<\/sub> \u2212 a<sub>n<\/sub> = 3\n<\/p>\n\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_69e22b4552e4b\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b4552e4b\"\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<p>\nDany jest ci\u0105g okre\u015blony wzorem:<br>\na<sub>n<\/sub> = 3n \u2212 1\n<\/p>\n\n<p>\nAby sprawdzi\u0107, czy ci\u0105g jest rosn\u0105cy, malej\u0105cy czy sta\u0142y, obliczamy r\u00f3\u017cnic\u0119 mi\u0119dzy kolejnymi wyrazami:\n<br><br>\na<sub>n+1<\/sub> \u2212 a<sub>n<\/sub>\n<\/p>\n\n<p>\nPodstawiamy wz\u00f3r:\n<br>\na<sub>n+1<\/sub> = 3(n + 1) \u2212 1 = 3n + 3 \u2212 1 = 3n + 2<br>\na<sub>n<\/sub> = 3n \u2212 1\n<\/p>\n\n<p>\nObliczamy r\u00f3\u017cnic\u0119:\n<br>\n(3n + 2) \u2212 (3n \u2212 1) = 3\n<\/p>\n\n<p>\nR\u00f3\u017cnica mi\u0119dzy kolejnymi wyrazami jest r\u00f3wna 3, czyli jest dodatnia.\n<\/p>\n\n<p>\nTo oznacza, \u017ce ka\u017cdy kolejny wyraz jest wi\u0119kszy od poprzedniego, wi\u0119c ci\u0105g jest <strong>rosn\u0105cy<\/strong>.\n<\/p>\n\n<p>\nDlatego poprawne odpowiedzi to:<br>\n<strong>a) rosn\u0105cy<\/strong><br>\n<strong>3) a<sub>n+1<\/sub> \u2212 a<sub>n<\/sub> = 3<\/strong>\n<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n\n\n\n<p><strong>zadanie 16.2.<\/strong> Najmniejsz\u0105 warto\u015bci\u0105 n , dla kt\u00f3rej wyraz (a<sub>n<\/sub>) jest wi\u0119kszy od 25 , jest:<\/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<p>a) 8<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) 9<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) 7<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>d) 26<\/p>\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_69e22b4552e4f\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b4552e4f\"\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) 9.\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_69e22b4552e52\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b4552e52\"\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<p>\nDany jest ci\u0105g okre\u015blony wzorem:<br>\na<sub>n<\/sub> = 3n \u2212 1\n<\/p>\n\n<p>\nMamy znale\u017a\u0107 najmniejsz\u0105 liczb\u0119 naturaln\u0105 n, dla kt\u00f3rej wyraz ci\u0105gu jest wi\u0119kszy ni\u017c 25, czyli spe\u0142nia warunek:<br>\na<sub>n<\/sub> > 25\n<\/p>\n\n<p>\nPodstawiamy wz\u00f3r na a<sub>n<\/sub>:\n<br>\n3n \u2212 1 > 25\n<\/p>\n\n<p>\nDodajemy 1 do obu stron:\n<br>\n3n > 26\n<\/p>\n\n<p>\nDzielimy przez 3:\n<br>\nn > 26\/3\n<\/p>\n\n<p>\n26\/3 \u2248 8,67\n<\/p>\n\n<p>\nNajmniejsza liczba naturalna wi\u0119ks\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n\n\n\n<p><strong>zadanie 16.3.<\/strong> Suma n pocz\u0105tkowych wyraz\u00f3w ci\u0105gu a<sub>n<\/sub> jest r\u00f3wna 57 dla n r\u00f3wnego:<\/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<p>a) 6<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) 23<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) 5<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>d) 11<\/p>\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_69e22b4552e54\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b4552e54\"\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) 6.\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_69e22b4552e57\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b4552e57\"\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<p>\nDany jest ci\u0105g okre\u015blony wzorem:<br>\na<sub>n<\/sub> = 3n \u2212 1\n<\/p>\n\n<p>\nMamy znale\u017a\u0107 takie n, dla kt\u00f3rego suma n pocz\u0105tkowych wyraz\u00f3w ci\u0105gu jest r\u00f3wna 57, czyli:<br>\nS<sub>n<\/sub> = 57\n<\/p>\n\n<p>\nKorzystamy ze wzoru na sum\u0119 n pocz\u0105tkowych wyraz\u00f3w ci\u0105gu arytmetycznego:<br>\nS<sub>n<\/sub> = (n\/2) \u00b7 (a<sub>1<\/sub> + a<sub>n<\/sub>)\n<\/p>\n\n<p>\nNajpierw obliczamy pierwszy wyraz:<br>\na<sub>1<\/sub> = 3\u00b71 \u2212 1 = 2\n<\/p>\n\n<p>\nNast\u0119pnie obliczamy a<sub>n<\/sub>:<br>\na<sub>n<\/sub> = 3n \u2212 1\n<\/p>\n\n<p>\nPodstawiamy do wzoru na sum\u0119:<br>\nS<sub>n<\/sub> = (n\/2) \u00b7 (2 + (3n \u2212 1))<br>\nS<sub>n<\/sub> = (n\/2) \u00b7 (3n + 1)\n<\/p>\n\n<p>\nWarunek zadania:<br>\n(n\/2)(3n + 1) = 57\n<\/p>\n\n<p>\nMno\u017cymy obie strony przez 2:<br>\nn(3n + 1) = 114\n<\/p>\n\n<p>\nRozwijamy nawias:<br>\n3n\u00b2 + n \u2212 114 = 0\n<\/p>\n\n<p>\nRozwi\u0105zujemy r\u00f3wnanie kwadratowe. Obliczamy delt\u0119:<br>\n\u0394 = 1\u00b2 + 4\u00b73\u00b7114 = 1369 = 37\u00b2\n<\/p>\n\n<p>\nn = (\u22121 + 37) \/ 6 = 36 \/ 6 = 6\n<\/p>\n\n<p>\nDrugi pierwiastek by\u0142by ujemny, wi\u0119c go odrzucamy.\n<\/p>\n\n<p>\n<strong>Wniosek:<\/strong> suma 57 pojawia si\u0119 dla n = 6.\n<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 16 \u2013 16.3 \u2013 wrzesie\u0144 2022 (3pkt) Dany jest ci\u0105g (an) okre\u015blony wzorem an=3n\u22121 dla ka\u017cdej liczby naturalnej n\u22651. zadanie 16.1. Doko\u0144cz zdanie tak, aby by\u0142o prawdziwe. Wybierz odpowied\u017a a, b albo c oraz jej uzasadnienie 1., 2. albo 3. Ci\u0105g an jest: a) rosn\u0105cy b) malej\u0105cy c) sta\u0142y Poniewa\u017c dla ka\u017cdej liczny naturalnej [&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":[25,204,136,81,83],"tags":[23,205,135],"class_list":["post-1348","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-rozne","category-wrzesien-2022","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-rozne","tag-wrzesien-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 16 \u2013 16.3 \u2013 wrzesie\u0144 2022 (3pkt) Dany jest ci\u0105g (an) okre\u015blony wzorem an=3n\u22121 dla ka\u017cdej liczby naturalnej n\u22651. zadanie 16.1. Doko\u0144cz zdanie tak, aby by\u0142o prawdziwe. Wybierz odpowied\u017a a, b albo c oraz jej uzasadnienie 1., 2. albo 3. Ci\u0105g an jest: a) rosn\u0105cy b) malej\u0105cy c) sta\u0142y Poniewa\u017c dla ka\u017cdej liczny naturalnej&hellip;","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1348","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=1348"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1348\/revisions"}],"predecessor-version":[{"id":1349,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1348\/revisions\/1349"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1348"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1348"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}