{"id":1279,"date":"2025-12-22T12:50:01","date_gmt":"2025-12-22T11:50:01","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1279"},"modified":"2025-12-22T12:50:06","modified_gmt":"2025-12-22T11:50:06","slug":"zadnie-115","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1279","title":{"rendered":"Zadnie 115"},"content":{"rendered":"\n<p><strong>zadanie 12 \u2013 czerwiec 2014 (1pkt)<\/strong><\/p>\n\n\n\n<p>W ci\u0105gu arytmetycznym (a<sub>n<\/sub>) , okre\u015blonym dla n\u22651 , dane s\u0105 dwa wyrazy: a<sub>2<\/sub>=11 i a<sub>4<\/sub>=7 . Suma czterech pocz\u0105tkowych wyraz\u00f3w 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) 36<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) 40<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) 13<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>d) 20<\/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_69e23d3e36d6b\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e23d3e36d6b\"\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) 40.\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_69e23d3e36d79\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e23d3e36d79\"\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 dwa wyrazy ci\u0105gu arytmetycznego:<\/p>\n\n<ul>\n  <li>a<sub>2<\/sub> = 11<\/li>\n  <li>a<sub>4<\/sub> = 7<\/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<h4>1. Wyznaczamy r\u00f3\u017cnic\u0119 r<\/h4>\n\n<p>Dla a<sub>2<\/sub> i a<sub>4<\/sub> zapisujemy:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>2<\/sub> = a<sub>1<\/sub> + r = 11\n<\/p>\n\n<p style=\"text-align:center;\">\na<sub>4<\/sub> = a<sub>1<\/sub> + 3r = 7\n<\/p>\n\n<p>Odejmujemy oba r\u00f3wnania:<\/p>\n\n<p style=\"text-align:center;\">\n(a<sub>1<\/sub> + 3r) \u2212 (a<sub>1<\/sub> + r) = 7 \u2212 11\n<\/p>\n\n<p style=\"text-align:center;\">\n2r = \u22124\n<\/p>\n\n<p style=\"text-align:center;\">\nr = \u22122\n<\/p>\n\n<h4>2. Wyznaczamy pierwszy wyraz a\u2081<\/h4>\n\n<p style=\"text-align:center;\">\na<sub>1<\/sub> + r = 11\n<\/p>\n\n<p style=\"text-align:center;\">\na<sub>1<\/sub> \u2212 2 = 11\n<\/p>\n\n<p style=\"text-align:center;\">\na<sub>1<\/sub> = 13\n<\/p>\n\n<h4>3. Obliczamy sum\u0119 czterech pierwszych wyraz\u00f3w<\/h4>\n\n<p>Wyznaczamy kolejne wyrazy:<\/p>\n\n<ul>\n  <li>a<sub>1<\/sub> = 13<\/li>\n  <li>a<sub>2<\/sub> = 11<\/li>\n  <li>a<sub>3<\/sub> = 9<\/li>\n  <li>a<sub>4<\/sub>\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 12 \u2013 czerwiec 2014 (1pkt) W ci\u0105gu arytmetycznym (an) , okre\u015blonym dla n\u22651 , dane s\u0105 dwa wyrazy: a2=11 i a4=7 . Suma czterech pocz\u0105tkowych wyraz\u00f3w tego ci\u0105gu jest r\u00f3wna: a) 36 b) 40 c) 13 d) 20<\/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":[25,141,195,81,83],"tags":[23,140,131],"class_list":["post-1279","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-czerwiec-2014","category-suma-poczatkowych-wyrazow-ciagu-ciag-arytmetyczny","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-czerwiec-2014","tag-suma-poczatkowych-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 12 \u2013 czerwiec 2014 (1pkt) W ci\u0105gu arytmetycznym (an) , okre\u015blonym dla n\u22651 , dane s\u0105 dwa wyrazy: a2=11 i a4=7 . Suma czterech pocz\u0105tkowych wyraz\u00f3w tego ci\u0105gu jest r\u00f3wna: a) 36 b) 40 c) 13 d) 20","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1279","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=1279"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1279\/revisions"}],"predecessor-version":[{"id":1281,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1279\/revisions\/1281"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1279"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1279"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}