{"id":10276,"date":"2024-11-05T11:05:33","date_gmt":"2024-11-05T10:05:33","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=10276"},"modified":"2024-11-05T11:05:36","modified_gmt":"2024-11-05T10:05:36","slug":"une-somme-de-racines-carrees","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/05\/une-somme-de-racines-carrees\/","title":{"rendered":"Une somme de racines carr\u00e9es"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Voici un exercice fort int\u00e9ressant qui demande de calculer la somme de racines carr\u00e9es: d\u00e9terminer la valeur de$$S=\\sum_{k=1}^{323} E(\\sqrt{k}),$$o\u00f9 \\(E(x)\\) d\u00e9signe la partie enti\u00e8re de \\(x\\).<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 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\/2024\/11\/05\/une-somme-de-racines-carrees\/#Vers_la_somme_des_racines_carrees_etude_preliminaire\" >Vers la somme des racines carr\u00e9es: \u00e9tude pr\u00e9liminaire<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/05\/une-somme-de-racines-carrees\/#Vers_la_somme_des_racines_carrees_nombre_de_termes_identiques\" >Vers la somme des racines carr\u00e9es: nombre de termes identiques<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/05\/une-somme-de-racines-carrees\/#Calcul_de_la_somme_des_racines_carrees\" >Calcul de la somme des racines carr\u00e9es<\/a><\/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\/2024\/11\/05\/une-somme-de-racines-carrees\/#Generalisation\" >G\u00e9n\u00e9ralisation<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Vers_la_somme_des_racines_carrees_etude_preliminaire\"><\/span>Vers la somme des racines carr\u00e9es: \u00e9tude pr\u00e9liminaire<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Regardons ce qui se passe pour les premiers termes de cette somme:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\" style=\"border-width:1px\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Valeurs de <em>k<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\">Valeurs de E(<em>\u221ak<\/em>)<\/th><\/tr><\/thead><tbody><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\">1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8230;<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8230;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">323<\/td><td class=\"has-text-align-center\" data-align=\"center\">17<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Notre somme ressemble donc \u00e0 la suivante :$$S = 1+1+2+2+2+3+3+\\cdots+17+17.$$ Il ne reste plus qu&rsquo;\u00e0 trouver le nombre de termes identiques.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Vers_la_somme_des_racines_carrees_nombre_de_termes_identiques\"><\/span>Vers la somme des racines carr\u00e9es: nombre de termes identiques<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Quel que soit la valeur de l&rsquo;entier <em>k<\/em>, $$(k+1)^2-k^2=2k+1.$$Ainsi, <\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>entre 1\u00b2 et 2\u00b2, il y a 2\u00d71 + 1 = 3 nombres entiers;<\/li>\n\n\n\n<li>entre 2\u00b2 et 3\u00b2, il y a 2\u00d72 + 1 = 5 nombres entiers;<\/li>\n\n\n\n<li>etc.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Notre somme peut alors s&rsquo;\u00e9crire:$$S = \\sum_{k=1}^{17} k \\times (2k+1).$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Calcul_de_la_somme_des_racines_carrees\"><\/span>Calcul de la somme des racines carr\u00e9es<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Rappelons les formules suivantes:$$\\sum_{k=1}^n k = \\frac{n(n+1)}{2}$$et$$\\sum_{k=1}^n k^2 = \\frac{n(n+1)(2n+1)}{6}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut alors \u00e9crire:$$\\begin{align}S &amp; =  \\sum_{k=1}^{17} k \\times (2k+1) \\\\ &amp;  =  2\\sum_{k=1}^{17} k^2 +  \\sum_{k=1}^{17} k\\\\ &amp; = 2 \\times \\frac{17 \\times 18 \\times 35}{6} + \\frac{17 \\times 18}{2}\\\\ &amp; = 3723. \\end{align}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si l&rsquo;on est peu s\u00fbr de soit, on peut v\u00e9rifier en Python:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>>>> sum( &#91; int(i**0.5) for i in range(1,324) ] )\n3723<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Je rappelle que \u00a0\u00bb i**0.5 \u00a0\u00bb d\u00e9signe la racine carr\u00e9e de i (si on ne souhaite pas importer la fonction \u00ab\u00a0sqrt\u00a0\u00bb du module \u00ab\u00a0math\u00a0\u00bb.<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Generalisation\"><\/span>G\u00e9n\u00e9ralisation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant que nous avons compris le fonctionnement d&rsquo;une telle somme, rien ne nous emp\u00eache de d\u00e9terminer l&rsquo;expression de la somme: $$S=\\sum_{k=1}^{n^2-1}E\\big( \\sqrt{k} \\big).$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En effet, sur le mod\u00e8le de ce qui a \u00e9t\u00e9 fait pr\u00e9c\u00e9demment, on peut \u00e9crire que:$$\\begin{align} S &amp; = \\sum_{k=1}^{n-1} k(2k+1) \\\\ &amp; = 2\\sum_{k=1}^{n-1} k^2 + \\sum_{k=1}^{n-1} k \\\\ &amp; = \\frac{n(n-1)(2n-1)}{3} + \\frac{n(n-1)}{2}\\\\&amp;=\\frac{n(n-1)(4n+1)}{6}. \\end{align}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nous venons alors de d\u00e9montrer la magnifique formule:$$\\boxed{ \\sum_{k=1}^{n^2-1}E\\big( \\sqrt{k} \\big) = \\frac{n(n-1)(4n+1)}{6}}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si vous \u00eates \u00e0 la recherche de cours particuliers de maths par webcam, n&rsquo;h\u00e9sitez pas \u00e0 faire appel \u00e0 moi et \u00e0 visiter mon site internet d\u00e9di\u00e9 \u00e0 cette activit\u00e9: <a href=\"https:\/\/courspasquet.fr\" target=\"_blank\" rel=\"noreferrer noopener\">courspasquet.fr<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Voici un exercice fort int\u00e9ressant qui demande de calculer la somme de racines carr\u00e9es: d\u00e9terminer la valeur de$$S=\\sum_{k=1}^{323} E(\\sqrt{k}),$$o\u00f9 \\(E(x)\\) d\u00e9signe la partie enti\u00e8re de \\(x\\).<\/p>\n","protected":false},"author":1,"featured_media":10284,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[],"class_list":["post-10276","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.1 - 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