{"id":1605,"date":"2025-12-31T13:24:56","date_gmt":"2025-12-31T12:24:56","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1605"},"modified":"2025-12-31T13:57:20","modified_gmt":"2025-12-31T12:57:20","slug":"zadanie-155","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1605","title":{"rendered":"Zadanie 155"},"content":{"rendered":"\n<p><strong>zadanie 26 &#8211; grudzie\u0144 2013 (zadanie otwarte) (2pkt)<\/strong><\/p>\n\n\n\n<p>Kwadrat K<sub>1<\/sub> ma bok d\u0142ugo\u015bci a. Obok niego rysujemy kolejno kwadraty K<sub>2<\/sub>, K<sub>3<\/sub>, K<sub>4<\/sub>,\u2026 takie, \u017ce ka\u017cdy kolejny kwadrat ma bok o po\u0142ow\u0119 mniejsz\u0105 od boku poprzedniego kwadratu (zobacz obrazek). Wyznacz pole kwadratu K<sub>12<\/sub>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"342\" height=\"212\" src=\"http:\/\/mymatma.pl\/wp-content\/uploads\/2025\/12\/Zrzut_ekranu_2025-12-31_131127-removebg-preview-1.png\" alt=\"\" class=\"wp-image-1609\" srcset=\"https:\/\/mymatma.pl\/wp-content\/uploads\/2025\/12\/Zrzut_ekranu_2025-12-31_131127-removebg-preview-1.png 342w, https:\/\/mymatma.pl\/wp-content\/uploads\/2025\/12\/Zrzut_ekranu_2025-12-31_131127-removebg-preview-1-300x186.png 300w\" sizes=\"(max-width: 342px) 100vw, 342px\" \/><\/figure>\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_69e238ef3e943\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e238ef3e943\"\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            \n<p>\nPole kwadratu <strong>K<sub>12<\/sub><\/strong> wynosi:\n<\/p>\n<p>\n<strong>\nP<sub>12<\/sub> = a<sup>2<\/sup> \/ 4<sup>11<\/sup>\n<\/strong>\n<\/p>\n\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_69e238ef3e94f\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e238ef3e94f\"\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>\nD\u0142ugo\u015bci bok\u00f3w kolejnych kwadrat\u00f3w tworz\u0105 ci\u0105g geometryczny.\nPierwszy kwadrat <strong>K<sub>1<\/sub><\/strong> ma bok d\u0142ugo\u015bci <strong>a<\/strong>,\na ka\u017cdy nast\u0119pny kwadrat ma bok o po\u0142ow\u0119 kr\u00f3tszy od poprzedniego.\n<\/p>\n\n<p>\nZatem d\u0142ugo\u015b\u0107 boku kwadratu <strong>K<sub>n<\/sub><\/strong> wynosi:\n<\/p>\n\n<p>\n<strong>\na<sub>n<\/sub> = a &middot; (1\/2)<sup>n-1<\/sup>\n<\/strong>\n<\/p>\n\n<p>\nPole kwadratu jest r\u00f3wne kwadratowi d\u0142ugo\u015bci jego boku, wi\u0119c:\n<\/p>\n\n<p>\n<strong>\nP<sub>n<\/sub> = a<sup>2<\/sup> &middot; (1\/4)<sup>n-1<\/sup>\n<\/strong>\n<\/p>\n\n<p>\nDla <strong>n = 12<\/strong> otrzymujemy:\n<\/p>\n\n<p>\n<strong>\nP<sub>12<\/sub> = a<sup>2<\/sup> &middot; (1\/4)<sup>11<\/sup> = a<sup>2<\/sup> \/ 4<sup>11<\/sup>\n<\/strong>\n<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 26 &#8211; grudzie\u0144 2013 (zadanie otwarte) (2pkt) Kwadrat K1 ma bok d\u0142ugo\u015bci a. Obok niego rysujemy kolejno kwadraty K2, K3, K4,\u2026 takie, \u017ce ka\u017cdy kolejny kwadrat ma bok o po\u0142ow\u0119 mniejsz\u0105 od boku poprzedniego kwadratu (zobacz obrazek). Wyznacz pole kwadratu K12.<\/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":[31,40,32,81,83],"tags":[29,39,28],"class_list":["post-1605","post","type-post","status-publish","format-standard","hentry","category-ciag-geometryczny","category-grudzien-2013","category-obliczanie-wyrazow-ciagu-geometrycznego","category-zadania-maturalne","category-zadania-tematami","tag-ciag-geometryczny","tag-grudzien-2013","tag-obliczanie-wyrazow-ciagu-geometrycznego"],"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 26 &#8211; grudzie\u0144 2013 (zadanie otwarte) (2pkt) Kwadrat K1 ma bok d\u0142ugo\u015bci a. Obok niego rysujemy kolejno kwadraty K2, K3, K4,\u2026 takie, \u017ce ka\u017cdy kolejny kwadrat ma bok o po\u0142ow\u0119 mniejsz\u0105 od boku poprzedniego kwadratu (zobacz obrazek). Wyznacz pole kwadratu K12.","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1605","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=1605"}],"version-history":[{"count":1,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1605\/revisions"}],"predecessor-version":[{"id":1610,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1605\/revisions\/1610"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1605"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1605"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1605"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}