{"id":1173,"date":"2025-12-18T14:41:04","date_gmt":"2025-12-18T13:41:04","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1173"},"modified":"2025-12-18T14:41:08","modified_gmt":"2025-12-18T13:41:08","slug":"zadanie-91","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1173","title":{"rendered":"Zadanie 91"},"content":{"rendered":"\n<p><strong>zadanie 14 \u2013 czerwiec 2015 (1 pkt)<\/strong><\/p>\n\n\n\n<p>Suma pierwszego i sz\u00f3stego wyrazu pewnego ci\u0105gu arytmetycznego jest r\u00f3wna 13. Wynika st\u0105d, \u017ce suma trzeciego i czwartego wyrazu tego ci\u0105gu jest r\u00f3wna:<\/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) 13<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) 12<\/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) 6<\/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_69e22b47d5019\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b47d5019\"\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) 13.\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_69e22b47d5025\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b47d5025\"\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>\n  Oznaczmy pierwszy wyraz ci\u0105gu jako <em>a\u2081<\/em>, a r\u00f3\u017cnic\u0119 ci\u0105gu jako <em>r<\/em>.\n<\/p>\n\n<p>\n  Z tre\u015bci zadania wiemy, \u017ce:\n  <br>\n  a\u2081 + a\u2086 = 13\n<\/p>\n\n<p>\n  Korzystamy ze wzoru na n-ty wyraz ci\u0105gu arytmetycznego:\n  <br>\n  a\u2086 = a\u2081 + 5r\n<\/p>\n\n<p>\n  Podstawiamy do r\u00f3wnania:\n  <br>\n  a\u2081 + (a\u2081 + 5r) = 13\n  <br>\n  2a\u2081 + 5r = 13\n<\/p>\n\n<p>\n  Teraz obliczamy sum\u0119 trzeciego i czwartego wyrazu:\n  <br>\n  a\u2083 = a\u2081 + 2r\n  <br>\n  a\u2084 = a\u2081 + 3r\n<\/p>\n\n<p>\n  Dodajemy:\n  <br>\n  a\u2083 + a\u2084 = (a\u2081 + 2r) + (a\u2081 + 3r) = 2a\u2081 + 5r\n<\/p>\n\n<p>\n  Z wcze\u015bniejszego r\u00f3wnania wiemy, \u017ce:\n  <br>\n  2a\u2081 + 5r = 13\n<\/p>\n\n<p>\n  Zatem suma trzeciego i czwartego wyrazu wynosi <strong>13<\/strong>.\n<\/p>\n\n<p>\n  Odpowied\u017a: <strong>a)<\/strong>\n<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 14 \u2013 czerwiec 2015 (1 pkt) Suma pierwszego i sz\u00f3stego wyrazu pewnego ci\u0105gu arytmetycznego jest r\u00f3wna 13. Wynika st\u0105d, \u017ce suma trzeciego i czwartego wyrazu tego ci\u0105gu jest r\u00f3wna: a) 13 b) 12 c) 7 d) 6<\/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":[25,98,26,81,83],"tags":[23,97,24],"class_list":["post-1173","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-czerwiec-2015","category-obliczanie-wyrazow-ciagu","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-czerwiec-2015","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 14 \u2013 czerwiec 2015 (1 pkt) Suma pierwszego i sz\u00f3stego wyrazu pewnego ci\u0105gu arytmetycznego jest r\u00f3wna 13. Wynika st\u0105d, \u017ce suma trzeciego i czwartego wyrazu tego ci\u0105gu jest r\u00f3wna: a) 13 b) 12 c) 7 d) 6","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1173","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=1173"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1173\/revisions"}],"predecessor-version":[{"id":1174,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1173\/revisions\/1174"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}