{"id":1195,"date":"2025-12-22T11:02:31","date_gmt":"2025-12-22T10:02:31","guid":{"rendered":"https:\/\/mymatma.pl\/?p=1195"},"modified":"2025-12-22T11:02:35","modified_gmt":"2025-12-22T10:02:35","slug":"zadanie-101","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1195","title":{"rendered":"Zadanie 101"},"content":{"rendered":"\n<p><strong>zadanie 13 \u2013 czerwiec 2017 (1pkt)<\/strong><\/p>\n\n\n\n<p>W ci\u0105gu arytmetycznym (a<sub>n<\/sub>) , okre\u015blonym dla n\u22651 , spe\u0142niony jest warunek 2a<sub>3<\/sub>=a<sub>2<\/sub>+a<sub>1<\/sub>+1 . R\u00f3\u017cnica r 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<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) 0<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) 1\/3<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) \u00bd<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>d) 1<\/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_69e22b9a87497\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b9a87497\"\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 b) 1\/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_69e22b9a874a3\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b9a874a3\"\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>Aby wyznaczy\u0107 r\u00f3\u017cnic\u0119 <strong>r<\/strong> ci\u0105gu arytmetycznego, skorzystajmy z definicji:<\/p>\n\n<ul>\n  <li>a\u2082 = a\u2081 + r<\/li>\n  <li>a\u2083 = a\u2081 + 2r<\/li>\n<\/ul>\n\n<p>W tre\u015bci zadania podano warunek:<\/p>\n\n<p style=\"text-align:center;\"><strong>2a\u2083 = a\u2082 + a\u2081 + 1<\/strong><\/p>\n\n<p>Podstawiamy wzory na wyrazy ci\u0105gu:<\/p>\n\n<p style=\"text-align:center;\">\n2(a\u2081 + 2r) = (a\u2081 + r) + a\u2081 + 1\n<\/p>\n\n<p>Upraszczamy obie strony r\u00f3wnania:<\/p>\n\n<p style=\"text-align:center;\">\n2a\u2081 + 4r = 2a\u2081 + r + 1\n<\/p>\n\n<p>Odejmujemy 2a\u2081 od obu stron:<\/p>\n\n<p style=\"text-align:center;\">\n4r = r + 1\n<\/p>\n\n<p>Przenosimy r na lew\u0105 stron\u0119:<\/p>\n\n<p style=\"text-align:center;\">\n3r = 1\n<\/p>\n\n<p>St\u0105d:<\/p>\n\n<p style=\"text-align:center; font-size: 1.2em;\">\n<strong>r = 1\/3<\/strong>\n<\/p>\n\n<p>Odpowied\u017a: <strong>b) 1\/3<\/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 13 \u2013 czerwiec 2017 (1pkt) W ci\u0105gu arytmetycznym (an) , okre\u015blonym dla n\u22651 , spe\u0142niony jest warunek 2a3=a2+a1+1 . R\u00f3\u017cnica r tego ci\u0105gu jest r\u00f3wna: a) 0 b) 1\/3 c) \u00bd d) 1<\/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":[1,25,127,160,81,83],"tags":[23,128,161],"class_list":["post-1195","post","type-post","status-publish","format-standard","hentry","category-bez-kategorii","category-ciag-arytmetyczny","category-czerwiec-2017","category-obliczanie-roznicy-ciagu","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-czerwiec-2017","tag-obliczanie-roznicy-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 13 \u2013 czerwiec 2017 (1pkt) W ci\u0105gu arytmetycznym (an) , okre\u015blonym dla n\u22651 , spe\u0142niony jest warunek 2a3=a2+a1+1 . R\u00f3\u017cnica r tego ci\u0105gu jest r\u00f3wna: a) 0 b) 1\/3 c) \u00bd d) 1","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1195","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=1195"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1195\/revisions"}],"predecessor-version":[{"id":1239,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1195\/revisions\/1239"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1195"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1195"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}