{"id":1272,"date":"2025-12-22T12:29:56","date_gmt":"2025-12-22T11:29:56","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1272"},"modified":"2025-12-22T12:30:00","modified_gmt":"2025-12-22T11:30:00","slug":"zadanie-112","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1272","title":{"rendered":"Zadanie 112"},"content":{"rendered":"\n<p><strong>zadanie 31 \u2013 sierpie\u0144 2016 ( zadanie otwarte) (4pkt)<\/strong><\/p>\n\n\n\n<p>Ci\u0105g arytmetyczny (a<sub>n<\/sub>) okre\u015blony jest wzorem a<sub>n<\/sub>=2016\u22123n , dla n\u22651 . Oblicz sum\u0119 wszystkich dodatnich wyraz\u00f3w 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_69e22b0cd8a05\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b0cd8a05\"\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            \nSuma wszystkich dodatnich wyraz\u00f3w tego ci\u0105gu to 6786.\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_69e22b0cd8a0f\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b0cd8a0f\"\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>Ci\u0105g arytmetyczny jest okre\u015blony wzorem:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>n<\/sub> = 2016 \u2212 3n\n<\/p>\n\n<p>Aby znale\u017a\u0107 sum\u0119 wszystkich dodatnich wyraz\u00f3w, najpierw ustalamy, dla jakich n wyraz jest dodatni:<\/p>\n\n<p style=\"text-align:center;\">\n2016 \u2212 3n > 0\n<\/p>\n\n<p>Przenosimy wyrazy:<\/p>\n\n<p style=\"text-align:center;\">\n2016 > 3n\n<\/p>\n\n<p>Dzielimy przez 3:<\/p>\n\n<p style=\"text-align:center;\">\nn < 672\n<\/p>\n\n<p>Oznacza to, \u017ce dodatnie s\u0105 wyrazy od n = 1 do n = 671.<\/p>\n\n<h4>Obliczamy pierwszy i ostatni dodatni wyraz:<\/h4>\n\n<ul>\n  <li>a<sub>1<\/sub> = 2016 \u2212 3\u00b71 = 2013<\/li>\n  <li>a<sub>671<\/sub> = 2016 \u2212 3\u00b7671 = 2016 \u2212 2013 = 3<\/li>\n<\/ul>\n\n<h4>Obliczamy sum\u0119 671 dodatnich wyraz\u00f3w:<\/h4>\n\n<p style=\"text-align:center;\">\nS<sub>671<\/sub> = (a<sub>1<\/sub> + a<sub>671<\/sub>) \u00b7 671 \/ 2\n<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>671<\/sub> = (2013 + 3) \u00b7 671 \/ 2\n<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>671<\/sub> = 2016 \u00b7 671 \/ 2\n<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>671<\/sub> = 1008 \u00b7 671 = 6786\n<\/p>\n\n<h3>Odpowied\u017a:<\/h3>\n<p>Suma wszystkich dodatnich wyraz\u00f3w ci\u0105gu wynosi <strong>6786<\/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 31 \u2013 sierpie\u0144 2016 ( zadanie otwarte) (4pkt) Ci\u0105g arytmetyczny (an) okre\u015blony jest wzorem an=2016\u22123n , dla n\u22651 . Oblicz sum\u0119 wszystkich dodatnich wyraz\u00f3w 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":[25,70,195,81,83],"tags":[23,69,131],"class_list":["post-1272","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-sierpien-2016","category-suma-poczatkowych-wyrazow-ciagu-ciag-arytmetyczny","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-sierpien-2016","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 31 \u2013 sierpie\u0144 2016 ( zadanie otwarte) (4pkt) Ci\u0105g arytmetyczny (an) okre\u015blony jest wzorem an=2016\u22123n , dla n\u22651 . Oblicz sum\u0119 wszystkich dodatnich wyraz\u00f3w tego ci\u0105gu.","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1272","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=1272"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1272\/revisions"}],"predecessor-version":[{"id":1273,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1272\/revisions\/1273"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1272"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1272"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1272"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}