{"id":1250,"date":"2025-12-22T11:47:55","date_gmt":"2025-12-22T10:47:55","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1250"},"modified":"2025-12-22T11:47:58","modified_gmt":"2025-12-22T10:47:58","slug":"zadanie-107","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1250","title":{"rendered":"Zadanie 107"},"content":{"rendered":"\n<p><strong>zadanie 15 &#8211; czerwiec 2025 (2pkt)<\/strong><\/p>\n\n\n\n<p>W ci\u0105gu arytmetycznym (a<sub>n<\/sub>) , okre\u015blonym dla ka\u017cdej liczby naturalnej n\u22651 , dane s\u0105 wyrazy: a<sub>1<\/sub>=52 oraz a<sub>25<\/sub>=2 .<\/p>\n\n\n\n<p><strong>zadanie 15.1.<\/strong> R\u00f3\u017cnica ci\u0105gu (a<sub>n<\/sub>) 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) -(25\/12)<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) -2<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) 2<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>d) 2\/12<\/p>\n<\/div>\n<\/div>\n\n\n\n<p><strong>Zadanie 15.2.<\/strong> Suma S<sub>25<\/sub> dwudziestu pi\u0119ciu pocz\u0105tkowych kolejnych wyraz\u00f3w ci\u0105gu (a<sub>n<\/sub>) 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) 675<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) 700<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) 1300<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>d) 1325<\/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_69e22b6c7835a\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22b6c7835a\"\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            \nZadanie 15.1:  \nR\u00f3\u017cnica ci\u0105gu: b) \u20132\n\nZadanie 15.2:  \nSuma S\u208d25\u208e: a) 675\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_69e22b6c78365\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22b6c78365\"\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<h4>Zadanie 15.1<\/h4>\n\n<p>W ci\u0105gu arytmetycznym obowi\u0105zuje wz\u00f3r:<\/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>W zadaniu podano:<\/p>\n\n<ul>\n  <li>a<sub>1<\/sub> = 52<\/li>\n  <li>a<sub>25<\/sub> = 2<\/li>\n<\/ul>\n\n<p>Podstawiamy do wzoru:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>25<\/sub> = a<sub>1<\/sub> + 24r\n<\/p>\n\n<p style=\"text-align:center;\">\n2 = 52 + 24r\n<\/p>\n\n<p>Odejmujemy 52 od obu stron:<\/p>\n\n<p style=\"text-align:center;\">\n24r = -50\n<\/p>\n\n<p style=\"text-align:center;\">\nr = -2\n<\/p>\n\n<h4>Zadanie 15.2<\/h4>\n\n<p>Korzystamy ze wzoru na sum\u0119 n pocz\u0105tkowych wyraz\u00f3w ci\u0105gu arytmetycznego:<\/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>Podstawiamy dane:<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>25<\/sub> = (52 + 2) \u00b7 25 \/ 2\n<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>25<\/sub> = 54 \u00b7 25 \/ 2\n<\/p>\n\n<p style=\"text-align:center;\">\nS<sub>25<\/sub> = 27 \u00b7 25 = 675\n<\/p>\n\n<h3>Odpowiedzi:<\/h3>\n<p>R\u00f3\u017cnica: r = -2<br>\nSuma: S<sub>25<\/sub> = 675<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 15 &#8211; czerwiec 2025 (2pkt) W ci\u0105gu arytmetycznym (an) , okre\u015blonym dla ka\u017cdej liczby naturalnej n\u22651 , dane s\u0105 wyrazy: a1=52 oraz a25=2 . zadanie 15.1. R\u00f3\u017cnica ci\u0105gu (an) jest r\u00f3wna: a) -(25\/12) b) -2 c) 2 d) 2\/12 Zadanie 15.2. Suma S25 dwudziestu pi\u0119ciu pocz\u0105tkowych kolejnych wyraz\u00f3w ci\u0105gu (an) jest r\u00f3wna: a) 675 [&hellip;]<\/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":[25,102,195,81,83],"tags":[23,100,131],"class_list":["post-1250","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-czerwiec-2025","category-suma-poczatkowych-wyrazow-ciagu-ciag-arytmetyczny","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-czerwiec-2025","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 15 &#8211; czerwiec 2025 (2pkt) W ci\u0105gu arytmetycznym (an) , okre\u015blonym dla ka\u017cdej liczby naturalnej n\u22651 , dane s\u0105 wyrazy: a1=52 oraz a25=2 . zadanie 15.1. R\u00f3\u017cnica ci\u0105gu (an) jest r\u00f3wna: a) -(25\/12) b) -2 c) 2 d) 2\/12 Zadanie 15.2. Suma S25 dwudziestu pi\u0119ciu pocz\u0105tkowych kolejnych wyraz\u00f3w ci\u0105gu (an) jest r\u00f3wna: a) 675&hellip;","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1250","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=1250"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1250\/revisions"}],"predecessor-version":[{"id":1251,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1250\/revisions\/1251"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1250"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1250"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1250"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}