{"id":1242,"date":"2025-12-22T11:13:13","date_gmt":"2025-12-22T10:13:13","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1242"},"modified":"2025-12-22T11:13:16","modified_gmt":"2025-12-22T10:13:16","slug":"zadanie-103","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1242","title":{"rendered":"Zadanie 103"},"content":{"rendered":"\n<p><strong>zadanie 31 \u2013 sierpie\u0144 2014 ( zadanie otwarte) (2pkt)<\/strong><\/p>\n\n\n\n<p>Dany jest ci\u0105g arytmetyczny (a<sub>n<\/sub>) okre\u015blony dla n\u22651 , w kt\u00f3rym a<sub>5<\/sub>=22 oraz a<sub>10<\/sub>=47 . Oblicz pierwszy wyraz a<sub>1<\/sub> i r\u00f3\u017cnic\u0119 r 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_69e22bff36976\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22bff36976\"\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\u2081 = -3, r = 5.\n        <\/div>\n    <\/div>\n    \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_69e22bff36981\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22bff36981\"\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>Rozwi\u0105zanie<\/h3>\n\n<p>W ci\u0105gu arytmetycznym znamy dwa wyrazy:<\/p>\n\n<ul>\n  <li>a<sub>5<\/sub> = 22<\/li>\n  <li>a<sub>10<\/sub> = 47<\/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<p>Podstawiamy dane:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>5<\/sub> = a<sub>1<\/sub> + 4r = 22\n<\/p>\n\n<p style=\"text-align:center;\">\na<sub>10<\/sub> = a<sub>1<\/sub> + 9r = 47\n<\/p>\n\n<p>Odejmujemy pierwsze r\u00f3wnanie od drugiego:<\/p>\n\n<p style=\"text-align:center;\">\n(a<sub>1<\/sub> + 9r) \u2212 (a<sub>1<\/sub> + 4r) = 47 \u2212 22\n<\/p>\n\n<p style=\"text-align:center;\">\n5r = 25\n<\/p>\n\n<p style=\"text-align:center;\">\nr = 5\n<\/p>\n\n<p>Podstawiamy r do r\u00f3wnania a<sub>1<\/sub> + 4r = 22:<\/p>\n\n<p style=\"text-align:center;\">\na<sub>1<\/sub> + 20 = 22\n<\/p>\n\n<p style=\"text-align:center;\">\na<sub>1<\/sub> = 2\n<\/p>\n\n<h3>Odpowied\u017a:<\/h3>\n<p>a<sub>1<\/sub> = 2 <br> r = 5<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 31 \u2013 sierpie\u0144 2014 ( zadanie otwarte) (2pkt) Dany jest ci\u0105g arytmetyczny (an) okre\u015blony dla n\u22651 , w kt\u00f3rym a5=22 oraz a10=47 . Oblicz pierwszy wyraz a1 i r\u00f3\u017cnic\u0119 r 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":[25,160,76,81,83],"tags":[23,161,75],"class_list":["post-1242","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-obliczanie-roznicy-ciagu","category-sierpien-2014","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-obliczanie-roznicy-ciagu","tag-sierpien-2014"],"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 2014 ( zadanie otwarte) (2pkt) Dany jest ci\u0105g arytmetyczny (an) okre\u015blony dla n\u22651 , w kt\u00f3rym a5=22 oraz a10=47 . Oblicz pierwszy wyraz a1 i r\u00f3\u017cnic\u0119 r tego ci\u0105gu.","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1242","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=1242"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1242\/revisions"}],"predecessor-version":[{"id":1243,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1242\/revisions\/1243"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1242"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1242"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1242"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}