{"id":1306,"date":"2025-12-22T13:34:20","date_gmt":"2025-12-22T12:34:20","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1306"},"modified":"2025-12-22T13:34:25","modified_gmt":"2025-12-22T12:34:25","slug":"zadanie-123","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1306","title":{"rendered":"Zadanie 123"},"content":{"rendered":"\n<p><strong>zadanie 30 \u2013 sierpie\u0144 2018 (zadanie otwarte) (2pkt)<\/strong><\/p>\n\n\n\n<p>Dziewi\u0105ty wyraz ci\u0105gu arytmetycznego (a<sub>n<\/sub>) , okre\u015blonego dla n\u22651 , jest r\u00f3wny 34 , a suma jego o\u015bmiu pocz\u0105tkowych wyraz\u00f3w jest r\u00f3wna 110 . Oblicz pierwszy wyraz i r\u00f3\u017cnic\u0119 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_69e22a5433005\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22a5433005\"\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 = 3.\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_69e22a5433011\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22a5433011\"\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 podano dwie informacje o ci\u0105gu arytmetycznym:<\/p>\n\n<ul>\n  <li>a<sub>9<\/sub> = 34<\/li>\n  <li>S<sub>8<\/sub> = 110<\/li>\n<\/ul>\n\n<p>Korzystamy ze wzoru na n\u2011ty wyraz ci\u0105gu arytmetycznego:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>n<\/sub> = a<sub>1<\/sub> + (n \u2212 1)r\n<\/p>\n\n<p>Dla n = 9:<\/p>\n\n<p style=\"text-align:center;\">\n34 = a<sub>1<\/sub> + 8r\n<\/p>\n\n<hr>\n\n<p>U\u017cywamy tak\u017ce wzoru na sum\u0119 n pocz\u0105tkowych wyraz\u00f3w:<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>n<\/sub> = (a<sub>1<\/sub> + a<sub>n<\/sub>) \u00b7 n \/ 2\n<\/p>\n\n<p>Dla n = 8:<\/p>\n\n<p style=\"text-align:center;\">\n110 = (a<sub>1<\/sub> + a<sub>8<\/sub>) \u00b7 4\n<\/p>\n\n<p>A poniewa\u017c:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>8<\/sub> = a<sub>1<\/sub> + 7r\n<\/p>\n\n<p>to podstawiamy:<\/p>\n\n<p style=\"text-align:center;\">\n110 = (a<sub>1<\/sub> + a<sub>1<\/sub> + 7r) \u00b7 4\n<\/p>\n\n<p style=\"text-align:center;\">\n110 = (2a<sub>1<\/sub> + 7r) \u00b7 4\n<\/p>\n\n<p>Dzielimy obie strony przez 4:<\/p>\n\n<p style=\"text-align:center;\">\n27.5 = 2a<sub>1<\/sub> + 7r\n<\/p>\n\n<hr>\n\n<h4>Rozwi\u0105zujemy uk\u0142ad r\u00f3wna\u0144:<\/h4>\n\n<p style=\"text-align:center;\">\n1) &nbsp; a<sub>1<\/sub> + 8r = 34  \n<br>\n2) &nbsp; 2a<sub>1<\/sub> + 7r = 27.5\n<\/p>\n\n<p>Po rozwi\u0105zaniu uk\u0142adu otrzymujemy:<\/p>\n\n<p style=\"text-align:center; font-size: 1.2em;\">\n<strong>a<sub>1<\/sub> = 3<\/strong>  \n<br>\n<strong>r = 4<\/strong>\n<\/p>\n\n<h3>Odpowied\u017a:<\/h3>\n<p>Pierwszy wyraz ci\u0105gu to <strong>3<\/strong>, a r\u00f3\u017cnica wynosi <strong>4<\/strong>.<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 30 \u2013 sierpie\u0144 2018 (zadanie otwarte) (2pkt) Dziewi\u0105ty wyraz ci\u0105gu arytmetycznego (an) , okre\u015blonego dla n\u22651 , jest r\u00f3wny 34 , a suma jego o\u015bmiu pocz\u0105tkowych wyraz\u00f3w jest r\u00f3wna 110 . Oblicz pierwszy wyraz i r\u00f3\u017cnic\u0119 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":[25,120,198,81,83],"tags":[23,119,197],"class_list":["post-1306","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-sierpien-2018","category-wlasnosci-trzech-kolejnych-wyrazow-ciagu","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-sierpien-2018","tag-wlasnosci-trzech-kolejnych-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 30 \u2013 sierpie\u0144 2018 (zadanie otwarte) (2pkt) Dziewi\u0105ty wyraz ci\u0105gu arytmetycznego (an) , okre\u015blonego dla n\u22651 , jest r\u00f3wny 34 , a suma jego o\u015bmiu pocz\u0105tkowych wyraz\u00f3w jest r\u00f3wna 110 . Oblicz pierwszy wyraz i r\u00f3\u017cnic\u0119 tego ci\u0105gu.","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1306","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=1306"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1306\/revisions"}],"predecessor-version":[{"id":1307,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1306\/revisions\/1307"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1306"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1306"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1306"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}