{"id":8116,"date":"2022-12-26T17:54:51","date_gmt":"2022-12-26T16:54:51","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=8116"},"modified":"2022-12-26T17:54:53","modified_gmt":"2022-12-26T16:54:53","slug":"suites-arithmetiques-de-degre-d","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/","title":{"rendered":"Suites arithm\u00e9tiques de degr\u00e9 d"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Vous connaissez s\u00fbrement les <a href=\"https:\/\/www.mathweb.fr\/euclide\/suites-arithmetiques-et-geometriques-avec-python\/\" target=\"_blank\" rel=\"noreferrer noopener\">suites arithm\u00e9tiques<\/a>. Mais savez-vous qu&#8217;elles font partie d&#8217;une plus grande famille ?<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_84 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#Suites_arithmetiques_de_degre_d_introduction\" >Suites arithm\u00e9tiques de degr\u00e9 d: introduction<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#La_suite_des_carres_parfaits\" >La suite des carr\u00e9s parfaits<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#La_suite_des_nombres_pentagonaux\" >La suite des nombres pentagonaux<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#Suites_arithmetiques_de_degre_d_generalites\" >Suites arithm\u00e9tiques de degr\u00e9 d: g\u00e9n\u00e9ralit\u00e9s<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#Definition_dune_suite_arithmetique_de_degre_d\" >D\u00e9finition d&#8217;une suite arithm\u00e9tique de degr\u00e9 d<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#Un_autre_exemple\" >Un autre exemple<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#Terme_general_dune_suite_arithmetique_de_degre_d\" >Terme g\u00e9n\u00e9ral d&#8217;une suite arithm\u00e9tique de degr\u00e9 d<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#Consequences\" >Cons\u00e9quences<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#Une_application_aux_suites_arithmetiques_de_degre_d_les_nombres_polygonaux\" >Une application aux suites arithm\u00e9tiques de degr\u00e9 d: les nombres polygonaux<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/12\/26\/suites-arithmetiques-de-degre-d\/#Un_peu_de_Python\" >Un peu de Python<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Suites_arithmetiques_de_degre_d_introduction\"><\/span>Suites arithm\u00e9tiques de degr\u00e9 <em>d<\/em>: introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_suite_des_carres_parfaits\"><\/span>La suite des carr\u00e9s parfaits<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons prendre l&#8217;exemple de la suite des carr\u00e9s parfaits:$$u_n=n^2$$.<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Valeurs de <em>n<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\">Valeurs de <em>n<\/em>\u00b2<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">16<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">25<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">36<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">49<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">64<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">81<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\">Premi\u00e8res valeurs de la suite d\u00e9finie par \\(u_n =n^2\\)<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, regardons la suite d\u00e9finie pour tout entier naturel <em>n<\/em> par:$$v_n=u_{n+1}-u_{n}.$$<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Valeurs de <em>n<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\">Valeurs de <em>n<\/em>\u00b2<\/th><th class=\"has-text-align-center\" data-align=\"center\">Valeurs de \\(v_n\\)<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(v_0 = 1 &#8211; 0 = 1\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\( v_1=4 &#8211; 1 = 3\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(v_2=9 &#8211; 4 = 5\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">16<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\( v_3=16 &#8211; 9 = 7\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">25<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(v_4=25 &#8211; 16 = 9\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">36<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(v_5=36 &#8211; 25 = 11\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">49<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(v_6=49 &#8211; 36 = 13\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">64<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(v_7=64 &#8211; 49 = 15\\)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">81<\/td><td class=\"has-text-align-center\" data-align=\"center\">\\(v_8=81 &#8211; 64 = 17\\)<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\">Premi\u00e8res valeurs de la suite d\u00e9finie par \\(v_n = u_n &#8211; u_{n-1}\\)<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">La suite \\( (v_n) \\) est arithm\u00e9tique de raison <em>r<\/em> = 2.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On dira alors que \\( (u_n) \\) est une <em>suite arithm\u00e9tique de degr\u00e9 2<\/em>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_suite_des_nombres_pentagonaux\"><\/span>La suite des nombres pentagonaux<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">On peut repr\u00e9senter cette suite \\( (u_n^{(0)}) \\) ainsi:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"557\" height=\"222\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image.png\" alt=\"suite arithm\u00e9tique de degr\u00e9 d: nombre pentagonaux\" class=\"wp-image-8118\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image.png 557w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-300x120.png 300w\" sizes=\"auto, (max-width: 557px) 100vw, 557px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">On peut &#8220;d\u00e9composer&#8221; cette suite ainsi:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Valeurs de \\(u_n^{(0)}\\)<\/th><th class=\"has-text-align-center\" data-align=\"center\">\\(u_n^{(1)} = u_n^{(0)} &#8211; u_{n-1}^{(0)}\\)<\/th><th class=\"has-text-align-center\" data-align=\"center\">\\(u_n^{(2)} = u_n^{(1)}-u_{n-1}^{(1)}\\)<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">\\(u_0=1\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\\(u_1=5\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\\(u_2=12\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\\(u_3=22\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">10<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\\(u_4=35\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">13<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Cette suite est aussi une suite arithm\u00e9tique de degr\u00e9 2. Dans ce tableau , j&#8217;ai volontairement &#8220;pouss\u00e9&#8221; jusqu&#8217;\u00e0 obtenir une suite constante pour mettre en relief la raison pour laquelle le degr\u00e9 est 2: on a &#8220;d\u00e9compos\u00e9&#8221; la suite initiale \u00e0 l&#8217;aide de 2 suites, dont la derni\u00e8re est constante.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Suites_arithmetiques_de_degre_d_generalites\"><\/span>Suites arithm\u00e9tiques de degr\u00e9 <em>d<\/em>: g\u00e9n\u00e9ralit\u00e9s<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Definition_dune_suite_arithmetique_de_degre_d\"><\/span>D\u00e9finition d&#8217;une suite arithm\u00e9tique de degr\u00e9 <em>d<\/em><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Une suite num\u00e9rique \\( (u_n^{(0)}) \\) est arithm\u00e9tique de degr\u00e9 <em>d<\/em> s&#8217;il existe un ensemble de suites \\( \\{ (u_n^{(1)}) \\}, \\{ (u_n^{(2)}) \\}, &#8230;, \\{ (u_n^{(d)}) \\} \\) telles que:$$\\forall p \\in [0 ; d-1]\\cap\\mathbb{N},\\ \\forall n\\in\\mathbb{N}^*,\\ u_n^{(p)} = u_{n-1}^{(p)}+u_{n-1}^{(p+1)}\\,,$$ o\u00f9 \\( (u_n^{(d)}) \\) est une suite constante.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La raison d&#8217;une telle suite est la valeur de \\( u_0^{(d)} \\).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_autre_exemple\"><\/span>Un autre exemple<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Si on d\u00e9coupe un plan \u00e0 l&#8217;aide de <em>n<\/em> droites, le nombre <strong><em>maximum <\/em><\/strong>de r\u00e9gions du plan est une suite arithm\u00e9tique de degr\u00e9 2 et de raison 1.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-1.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"912\" height=\"326\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-1.png\" alt=\"\" class=\"wp-image-8122\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-1.png 912w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-1-600x214.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-1-300x107.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-1-768x275.png 768w\" sizes=\"auto, (max-width: 912px) 100vw, 912px\" \/><\/a><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Terme_general_dune_suite_arithmetique_de_degre_d\"><\/span>Terme g\u00e9n\u00e9ral d&#8217;une suite arithm\u00e9tique de degr\u00e9 <em>d<\/em><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Si \\( (u_n^{(d)}) \\) est une suite arithm\u00e9tique de degr\u00e9 <em>d<\/em> et de raison <em>r<\/em>, alors:$$\\forall n\\in\\mathbb{N},\\ u_n^{(0)} = \\sum_{m=0}^{d-1}\\left(u_0^{(m)}P_m(n)\\right)+rP_d(n)\\,,$$o\u00f9 la famille de polyn\u00f4mes \\(P_k(n)\\) est d\u00e9finie par:$$\\forall n\\in\\mathbb{N},\\ \\begin{cases} P_0(n)=1\\\\\\forall k\\in\\mathbb{N},\\ P_{k+1}(n)=\\displaystyle\\sum_{j=1}^nP_k(j) \\end{cases}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La d\u00e9monstration est plut\u00f4t longue mais elle est disponible dans mon livre &#8220;<a href=\"https:\/\/www.mathweb.fr\/euclide\/ouvrages-personnels-de-mathematiques\/\">Ainsi de suite<\/a>&#8220;, disponible gratuitement en t\u00e9l\u00e9chargement au format PDF. Cela se d\u00e9montre par r\u00e9currence.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Consequences\"><\/span>Cons\u00e9quences<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Ce dernier r\u00e9sultat montre que le terme g\u00e9n\u00e9ral d&#8217;une suite arithm\u00e9tique de degr\u00e9 <em>d<\/em> s&#8217;exprime comme un polyn\u00f4me de degr\u00e9 <em>d<\/em>:$$\\forall\\  n\\in\\mathbb{N},\\ u_n^{(d)} = \\alpha_dn^d + \\cdots + \\alpha_1n + \\alpha_0.$$ Pour d\u00e9terminer la valeur des coefficients \u000b\\(\\alpha<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">_k\\), il suffit de r\u00e9soudre le syst\u00e8me lin\u00e9aire de Vandermonde suivant :$$\\begin{cases}u_1^{(0)} &amp; = &amp; \\alpha_0 + \\alpha_1 + \\cdots + \\alpha_d\\\\u_2^{(0)} &amp; = &amp; \\alpha_0 + 2\\alpha_1 + \\cdots + 2^d\\alpha_d\\\\\\vdots &amp; &amp; \\\\u_{d+1}^{(0)} &amp; = &amp; \\alpha_0 + (d+1)\\alpha_1 + \\cdots + (d+1)^d\\alpha_d\\end{cases} $$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_application_aux_suites_arithmetiques_de_degre_d_les_nombres_polygonaux\"><\/span>Une application aux suites arithm\u00e9tiques de degr\u00e9 <em>d<\/em>: les nombres polygonaux<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Nous en avons bri\u00e8vement parl\u00e9 pr\u00e9c\u00e9demment lorsque nous avons regard\u00e9 la suite pentagonale. <\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"648\" height=\"201\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-2.png\" alt=\"\" class=\"wp-image-8126\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-2.png 648w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-2-600x186.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-2-300x93.png 300w\" sizes=\"auto, (max-width: 648px) 100vw, 648px\" \/><figcaption class=\"wp-element-caption\">a. D\u00e9but de la suite \\( (u_n^{(3)}) \\)<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"656\" height=\"214\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-3.png\" alt=\"\" class=\"wp-image-8127\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-3.png 656w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-3-600x196.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-3-300x98.png 300w\" sizes=\"auto, (max-width: 656px) 100vw, 656px\" \/><figcaption class=\"wp-element-caption\">b. D\u00e9but de la suite \\( (u_n^{(4)}) \\)<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"706\" height=\"258\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-4.png\" alt=\"\" class=\"wp-image-8128\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-4.png 706w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-4-600x219.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/12\/image-4-300x110.png 300w\" sizes=\"auto, (max-width: 706px) 100vw, 706px\" \/><figcaption class=\"wp-element-caption\">c. D\u00e9but de la suite \\( (u_n^{(5)}) \\)<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">On arrive \u00e0 d\u00e9montrer \u00e0 l&#8217;aide du r\u00e9sultat pr\u00e9c\u00e9dent que:$$\\forall\\; p\\geqslant 3,\\ u_n^{(p)} = \\frac{n\\left(4-p+(p-2)n\\right)}{2}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_peu_de_Python\"><\/span>Un peu de Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Je vous propose un programme Python qui permet de voir si une suite est arithm\u00e9tique de degr\u00e9 <em>d<\/em>.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def is_ardeg(liste , degre = 0):\n    if len(liste) != 1:\n        comp = [0] * len(liste)\n        if liste == comp:\n            return True , degre-1\n        else:\n            # on v\u00e9rifie s'il y a deux \u00e9l\u00e9ments identiques dans la liste\n            comp = [ liste[0] ] * len(liste)\n            if liste != comp:\n                for e in liste:\n                    i = liste.index(e)\n                    L = liste[:i] + liste[i+1:]\n                    if e in L:\n                        return False\n                \n            u = []\n            for n in range(1,len(liste)):\n                u.append( liste[n-1] - liste[n] )\n\n            return is_ardeg(u,degre+1)\n    else:\n        return False , \"Pas assez d'\u00e9l\u00e9ments pour conclure.\"<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>>>> L = &#91;i**2 for i in range(10)]\n>>> is_ardeg(L)\n(True, 2)\n>>> is_ardeg(&#91;1, 3, 5, 7, 8, 9, 11])\nFalse<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Vous connaissez s\u00fbrement les suites arithm\u00e9tiques. Mais savez-vous qu&#8217;elles font partie d&#8217;une plus grande famille ?<\/p>\n","protected":false},"author":1,"featured_media":8130,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[377],"class_list":["post-8116","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python","tag-suites-arithmetiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.7 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Suites arithm\u00e9tiques de degr\u00e9 d - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Vous connaissez les suites arithm\u00e9tiques de degr\u00e9 d ? C&#039;est une g\u00e9n\u00e9ralit\u00e9 des suites arithm\u00e9tiques que l&#039;on \u00e9tudie au lyc\u00e9e. 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