{"id":1343,"date":"2025-12-23T14:18:05","date_gmt":"2025-12-23T13:18:05","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1343"},"modified":"2025-12-23T14:18:08","modified_gmt":"2025-12-23T13:18:08","slug":"zadanie-137","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1343","title":{"rendered":"Zadanie 137"},"content":{"rendered":"\n<p><strong>zadanie 15 \u2013 pokazowy 2023 (1pkt)<\/strong><\/p>\n\n\n\n<p>Dany jest ci\u0105g a<sub>n<\/sub> okre\u015blony wzorem a<sub>n<\/sub> = -3n + 5 dla ka\u017cdej liczby naturalnej n\u22651. Oce\u0144 prawdziwo\u015b\u0107 poni\u017cszych stwierdze\u0144. Wybierz <strong>P<\/strong>, je\u015bli stwierdzenie jest prawdziwe, albo <strong>F<\/strong> \u2013 je\u015bli jest fa\u0142szywe.<\/p>\n\n\n\n<p>Liczby 2,(-1),(-4) s\u0105 trzema kolejnymi pocz\u0105tkowymi wyrazami ci\u0105gu a<sub>n<\/sub>. <strong>P \/ F<\/strong><\/p>\n\n\n\n<p>a<sub>n<\/sub> jest ci\u0105giem arytmetycznym o r\u00f3\u017cnicy r\u00f3wnej 5. <strong>P \/ F<\/strong><\/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_69e22ba0e065c\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22ba0e065c\"\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\u2026.\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_69e22ba0e0667\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22ba0e0667\"\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>\nPan Stanis\u0142aw sp\u0142aci\u0142 po\u017cyczk\u0119 w wysoko\u015bci 8910 z\u0142 w 18 ratach, kt\u00f3re tworz\u0105 ci\u0105g arytmetyczny malej\u0105cy. \nOznacza to, \u017ce ka\u017cda kolejna rata jest mniejsza od poprzedniej o 30 z\u0142.\n<\/p>\n\n<p>\nNiech a<sub>1<\/sub> oznacza pierwsz\u0105 rat\u0119, a d = \u221230 r\u00f3\u017cnic\u0119 ci\u0105gu.\n<\/p>\n\n<p>\nSuma 18 rat to suma 18 wyraz\u00f3w ci\u0105gu arytmetycznego, wi\u0119c korzystamy ze wzoru:\n<br>\nS<sub>n<\/sub> = (n\/2) \u00b7 (2a<sub>1<\/sub> + (n \u2212 1)d)\n<\/p>\n\n<p>\nPodstawiamy dane:\n<br>\n8910 = (18\/2) \u00b7 (2a<sub>1<\/sub> + 17 \u00b7 (\u221230))\n<\/p>\n\n<p>\nUpraszczamy:\n<br>\n8910 = 9 \u00b7 (2a<sub>1<\/sub> \u2212 510)\n<\/p>\n\n<p>\nDzielimy obie strony przez 9:\n<br>\n990 = 2a<sub>1<\/sub> \u2212 510\n<\/p>\n\n<p>\nDodajemy 510:\n<br>\n1500 = 2a<sub>1<\/sub>\n<\/p>\n\n<p>\nDzielimy przez 2:\n<br>\na<sub>1<\/sub> = 750\n<\/p>\n\n<p>\n<strong>Wniosek:<\/strong> pierwsza rata wynosi 750 z\u0142.\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 15 \u2013 pokazowy 2023 (1pkt) Dany jest ci\u0105g an okre\u015blony wzorem an = -3n + 5 dla ka\u017cdej liczby naturalnej n\u22651. Oce\u0144 prawdziwo\u015b\u0107 poni\u017cszych stwierdze\u0144. Wybierz P, je\u015bli stwierdzenie jest prawdziwe, albo F \u2013 je\u015bli jest fa\u0142szywe. Liczby 2,(-1),(-4) s\u0105 trzema kolejnymi pocz\u0105tkowymi wyrazami ci\u0105gu an. P \/ F an jest ci\u0105giem arytmetycznym o [&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":[25,151,204,81,83],"tags":[23,150,205],"class_list":["post-1343","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-pokazowy-2023","category-rozne","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-pokazowy-2023","tag-rozne"],"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 pokazowy 2023 (1pkt) Dany jest ci\u0105g an okre\u015blony wzorem an = -3n + 5 dla ka\u017cdej liczby naturalnej n\u22651. Oce\u0144 prawdziwo\u015b\u0107 poni\u017cszych stwierdze\u0144. Wybierz P, je\u015bli stwierdzenie jest prawdziwe, albo F \u2013 je\u015bli jest fa\u0142szywe. Liczby 2,(-1),(-4) s\u0105 trzema kolejnymi pocz\u0105tkowymi wyrazami ci\u0105gu an. P \/ F an jest ci\u0105giem arytmetycznym o&hellip;","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1343","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=1343"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1343\/revisions"}],"predecessor-version":[{"id":1344,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1343\/revisions\/1344"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1343"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1343"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}