{"id":1133,"date":"2025-12-18T12:25:31","date_gmt":"2025-12-18T11:25:31","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1133"},"modified":"2025-12-18T12:35:24","modified_gmt":"2025-12-18T11:35:24","slug":"zadanie-84","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1133","title":{"rendered":"Zadanie 84"},"content":{"rendered":"\n<p><strong>zadanie 9 \u2013 czerwiec 2019 (1pkt)<\/strong><\/p>\n\n\n\n<p>Dany jest rosn\u0105cy ci\u0105g arytmetyczny (a<sub>n<\/sub>) , okre\u015blony dla liczb naturalnych n\u22651 , o wyrazach dodatnich. Je\u015bli a<sub>2<\/sub>+a<sub>9<\/sub>=a<sub>4<\/sub>+a<sub>k<\/sub> , to k jest r\u00f3wne:<\/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) 7<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) 6<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>d) 5<\/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_69e22b6f0440a\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b6f0440a\"\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            \n<b>a) 8<\/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_69e22b6f04415\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b6f04415\"\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>\nW zadaniu podano zale\u017cno\u015b\u0107 mi\u0119dzy wyrazami ci\u0105gu arytmetycznego:<br>\na\u2082 + a\u2089 = a\u2084 + a\u2096.\n<br><br>\nKorzystamy ze wzoru na n\u2011ty wyraz ci\u0105gu arytmetycznego:<br>\na\u2099 = a\u2081 + (n \u2212 1)r.\n<br><br>\nZapisujemy wyrazy:<br>\na\u2082 = a\u2081 + r<br>\na\u2084 = a\u2081 + 3r<br>\na\u2089 = a\u2081 + 8r<br>\na\u2096 = a\u2081 + (k \u2212 1)r\n<br><br>\nPodstawiamy do r\u00f3wnania:<br>\n(a\u2081 + r) + (a\u2081 + 8r) = (a\u2081 + 3r) + (a\u2081 + (k \u2212 1)r)\n<br><br>\nUpraszczenie obu stron daje:<br>\n2a\u2081 + 9r = 2a\u2081 + (k + 2)r\n<br><br>\nPor\u00f3wnujemy wsp\u00f3\u0142czynniki przy r:<br>\n9 = k + 2 \u2192 k = 7\n<br><br>\n<strong>Prawid\u0142owa odpowied\u017a: b) 7<\/strong>\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 9 \u2013 czerwiec 2019 (1pkt) Dany jest rosn\u0105cy ci\u0105g arytmetyczny (an) , okre\u015blony dla liczb naturalnych n\u22651 , o wyrazach dodatnich. Je\u015bli a2+a9=a4+ak , to k jest r\u00f3wne: a) 8 b) 7 c) 6 d) 5<\/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":[25,66,26,81,83],"tags":[23,65,24],"class_list":["post-1133","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-czerwiec-2019","category-obliczanie-wyrazow-ciagu","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-czerwiec-2019","tag-obliczanie-wyrazow-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 9 \u2013 czerwiec 2019 (1pkt) Dany jest rosn\u0105cy ci\u0105g arytmetyczny (an) , okre\u015blony dla liczb naturalnych n\u22651 , o wyrazach dodatnich. Je\u015bli a2+a9=a4+ak , to k jest r\u00f3wne: a) 8 b) 7 c) 6 d) 5","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1133","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=1133"}],"version-history":[{"count":2,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1133\/revisions"}],"predecessor-version":[{"id":1142,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1133\/revisions\/1142"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1133"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1133"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}