{"id":1287,"date":"2025-12-22T13:04:03","date_gmt":"2025-12-22T12:04:03","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1287"},"modified":"2025-12-22T13:04:08","modified_gmt":"2025-12-22T12:04:08","slug":"zadanie-118","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1287","title":{"rendered":"Zadanie 118"},"content":{"rendered":"\n<p><strong>zadanie 31 \u2013 zbi\u00f3r zada\u0144 CKE (zadanie otwarte) (2pkt)<\/strong><\/p>\n\n\n\n<p>Dany jest ci\u0105g arytmetyczny (an) okre\u015blony dla ka\u017cdej liczby naturalnej n\u22651 . Jego r\u00f3\u017cnica jest r\u00f3wna 4 , a suma jego pierwszych pi\u0119ciu wyraz\u00f3w jest trzy razy mniejsza od sumy nast\u0119pnych pi\u0119ciu wyraz\u00f3w. Oblicz pierwszy wyraz tego ci\u0105gu.<\/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_69e22b3f1f543\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b3f1f543\"\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\u2081 = -26.\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_69e22b3f1f54f\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b3f1f54f\"\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<h3>Wyja\u015bnienie<\/h3>\n\n<p>W zadaniu dany jest ci\u0105g arytmetyczny o r\u00f3\u017cnicy:<\/p>\n\n<p style=\"text-align:center;\">\nr = 4\n<\/p>\n\n<p>Wiemy, \u017ce suma pierwszych pi\u0119ciu wyraz\u00f3w jest trzy razy mniejsza od sumy kolejnych pi\u0119ciu wyraz\u00f3w:<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>5<\/sub> = 3 \u00b7 S<sub>6\u201310<\/sub>\n<\/p>\n\n<h4>1. Zapisujemy sumy wyraz\u00f3w<\/h4>\n\n<p>Pierwsze pi\u0119\u0107 wyraz\u00f3w:<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>5<\/sub> = (a<sub>1<\/sub> + a<sub>5<\/sub>) \u00b7 5 \/ 2\n<\/p>\n\n<p>Wyraz pi\u0105ty:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>5<\/sub> = a<sub>1<\/sub> + 4\u00b74 = a<sub>1<\/sub> + 16\n<\/p>\n\n<p>Zatem:<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>5<\/sub> = (a<sub>1<\/sub> + a<sub>1<\/sub> + 16) \u00b7 5 \/ 2 = (2a<sub>1<\/sub> + 16) \u00b7 5 \/ 2\n<\/p>\n\n<hr>\n\n<p>Suma wyraz\u00f3w od 6 do 10:<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>6\u201310<\/sub> = S<sub>10<\/sub> \u2212 S<sub>5<\/sub>\n<\/p>\n\n<p>Najpierw obliczamy S<sub>10<\/sub>:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>10<\/sub> = a<sub>1<\/sub> + 9\u00b74 = a<sub>1<\/sub> + 36\n<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>10<\/sub> = (a<sub>1<\/sub> + a<sub>10<\/sub>) \u00b7 10 \/ 2 = (a<sub>1<\/sub> + a<sub>1<\/sub> + 36) \u00b7 5 = (2a<sub>1<\/sub> + 36) \u00b7 5\n<\/p>\n\n<p>Zatem:<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>6\u201310<\/sub> = (2a<sub>1<\/sub> + 36)\u00b75 \u2212 (2a<sub>1<\/sub> + 16)\u00b75\/2\n<\/p>\n\n<h4>2. Korzystamy z warunku zadania<\/h4>\n\n<p style=\"text-align:center;\">\n(2a<sub>1<\/sub> + 16) \u00b7 5 \/ 2 = 3 \u00b7 S<sub>6\u201310<\/sub>\n<\/p>\n\n<p>Po uproszczeniu r\u00f3wnania otrzymujemy:<\/p>\n\n<p style=\"text-align:center; font-size: 1.2em;\">\na<sub>1<\/sub> = \u221226\n<\/p>\n\n<h3>Odpowied\u017a:<\/h3>\n<p>Pierwszy wyraz ci\u0105gu wynosi <strong>\u221226<\/strong>.<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 31 \u2013 zbi\u00f3r zada\u0144 CKE (zadanie otwarte) (2pkt) Dany jest ci\u0105g arytmetyczny (an) okre\u015blony dla ka\u017cdej liczby naturalnej n\u22651 . Jego r\u00f3\u017cnica jest r\u00f3wna 4 , a suma jego pierwszych pi\u0119ciu wyraz\u00f3w jest trzy razy mniejsza od sumy nast\u0119pnych pi\u0119ciu wyraz\u00f3w. Oblicz pierwszy wyraz tego ci\u0105gu.<\/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":[25,195,81,83,146],"tags":[],"class_list":["post-1287","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-suma-poczatkowych-wyrazow-ciagu-ciag-arytmetyczny","category-zadania-maturalne","category-zadania-tematami","category-zbior-zadan-cke"],"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 31 \u2013 zbi\u00f3r zada\u0144 CKE (zadanie otwarte) (2pkt) Dany jest ci\u0105g arytmetyczny (an) okre\u015blony dla ka\u017cdej liczby naturalnej n\u22651 . Jego r\u00f3\u017cnica jest r\u00f3wna 4 , a suma jego pierwszych pi\u0119ciu wyraz\u00f3w jest trzy razy mniejsza od sumy nast\u0119pnych pi\u0119ciu wyraz\u00f3w. Oblicz pierwszy wyraz tego ci\u0105gu.","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1287","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=1287"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1287\/revisions"}],"predecessor-version":[{"id":1289,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1287\/revisions\/1289"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1287"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1287"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1287"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}