{"id":4048,"date":"2025-02-13T09:57:46","date_gmt":"2025-02-13T08:57:46","guid":{"rendered":"https:\/\/jhmaths.fr\/?page_id=4048"},"modified":"2025-02-13T10:09:03","modified_gmt":"2025-02-13T09:09:03","slug":"suites-arithmetiques-et-geometriques","status":"publish","type":"page","link":"https:\/\/jhmaths.fr\/index.php\/premiere-generale\/suites-arithmetiques-et-geometriques\/","title":{"rendered":"Suites arithm\u00e9tiques et g\u00e9om\u00e9triques"},"content":{"rendered":"\n<h1 class=\"wp-block-heading\">Suites arithm\u00e9tiques<\/h1>\n\n\n\n<div class=\"wp-block-group propriete is-vertical is-layout-flex wp-container-core-group-is-layout-831b2db5 wp-block-group-is-layout-flex\">\n<p class=\"wp-block-paragraph\">Une suite arithm\u00e9tique de raison \\(r\\) est d\u00e9finie par :<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>la relation de r\u00e9currence $u_{n+1}=u_n+r$&nbsp;<\/li>\n\n\n\n<li>son expression explicite $u_n=u_0+nr$<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-group propriete is-vertical is-layout-flex wp-container-core-group-is-layout-831b2db5 wp-block-group-is-layout-flex\">\n<p class=\"wp-block-paragraph\"><strong>Repr\u00e9sentation graphique :<\/strong> les points sont align\u00e9s.<br>Ici, la repr\u00e9sentation de la suite de premier terme 2 et de raison 0,5 ($u_n=2+0,5n$)<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"777\" height=\"547\" src=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2025\/02\/illu-suite-arithmetique.png\" alt=\"\" class=\"wp-image-4049\" srcset=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2025\/02\/illu-suite-arithmetique.png 777w, https:\/\/jhmaths.fr\/wp-content\/uploads\/2025\/02\/illu-suite-arithmetique-300x211.png 300w, https:\/\/jhmaths.fr\/wp-content\/uploads\/2025\/02\/illu-suite-arithmetique-768x541.png 768w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/figure>\n<\/div>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9terminer l&#039;expression g\u00e9n\u00e9rale d&#039;une suite arithm\u00e9tique - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6O0KhPMHvBA?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Variations<\/h2>\n\n\n\n<p class=\"propriete wp-block-paragraph\">Si $r$ est positif, la suite est <strong>croissante <\/strong><br>Si $r$ est n\u00e9gatif, la suite est d\u00e9<strong>croissante <\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Somme des termes<\/h2>\n\n\n\n<p class=\"propriete wp-block-paragraph\"><strong>$1+2+\\dots+n=\\dfrac{n(n+1)}{2}$<\/strong><br><br>Et pour tout suite arithm\u00e9tique de raison $r$<strong> :<\/strong><br><br>$u_0+u_1+\\dots+u_n=(n+1)\\dfrac{u_0+u_n}{2}$<br><br>$u_r+u_{r+1}+\\dots+u_n= (n-r+1)\\dfrac{u_r+u_n}{2}$<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Calculer la somme des termes d&#039;une suite arithm\u00e9tique (1) - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/WeDtB9ZUTHs?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h1 class=\"wp-block-heading\">Suites g\u00e9om\u00e9triques<\/h1>\n\n\n\n<div class=\"wp-block-group propriete is-vertical is-layout-flex wp-container-core-group-is-layout-831b2db5 wp-block-group-is-layout-flex\">\n<p class=\"wp-block-paragraph\">Une suite g\u00e9om\u00e9trique de raison \\(q\\) est d\u00e9finie par :<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>la relation de r\u00e9currence $u_{n+1}=u_n\\times q$\u00a0<\/li>\n\n\n\n<li>son expression explicite $u_n=u_0\\times q^n$<\/li>\n<\/ul>\n<\/div>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9montrer qu&#039;une suite est g\u00e9om\u00e9trique - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/YPbEHxuMaeQ?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<div class=\"wp-block-group propriete is-vertical is-layout-flex wp-container-core-group-is-layout-831b2db5 wp-block-group-is-layout-flex\">\n<p class=\"wp-block-paragraph\"><strong>Repr\u00e9sentation graphique :<\/strong> les points ne sont pas align\u00e9s.<br>Ici, la repr\u00e9sentation de la suite de premier terme 2 et de raison 0,5 ($u_n=2\\times0,5^n$).<br>On verra par la suite que les points suivent une <strong>courbe exponentielle<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"776\" height=\"631\" src=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2025\/02\/illu-suite-geometrique.png\" alt=\"\" class=\"wp-image-4052\" srcset=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2025\/02\/illu-suite-geometrique.png 776w, https:\/\/jhmaths.fr\/wp-content\/uploads\/2025\/02\/illu-suite-geometrique-300x244.png 300w, https:\/\/jhmaths.fr\/wp-content\/uploads\/2025\/02\/illu-suite-geometrique-768x624.png 768w\" sizes=\"auto, (max-width: 776px) 100vw, 776px\" \/><\/figure>\n<\/div>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9terminer l&#039;expression g\u00e9n\u00e9rale d&#039;une suite g\u00e9om\u00e9trique - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/WTmdtbQpa0c?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Variations<\/h2>\n\n\n\n<p class=\"propriete wp-block-paragraph\">Pour une suite g\u00e9om\u00e9trique de premier terme $u_0$ positif :<br>Si $q>1$, la suite est <strong>croissante <\/strong><br>Si $0&lt;q&lt;1$, la suite est d\u00e9<strong>croissante <\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Somme des termes<\/h2>\n\n\n\n<p class=\"propriete wp-block-paragraph\"><strong>$1+q+q^2+\\dots+q^n=\\dfrac{q^n-1}{q-1}$<\/strong><br><br>Et pour tout suite g\u00e9om\u00e9trique de raison $q$<strong> :<\/strong><br><br>$u_0+u_1+\\dots+u_n=u_0\\dfrac{q^{n+1}-1}{q-1}$<br><br>$u_r+u_{r+1}+\\dots+u_n= u_r\\dfrac{q^{n-r+1}-1}{q-1}$<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Calculer la somme des termes d&#039;une suite g\u00e9om\u00e9trique (1) - Terminale Techno\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_BjEOTi-2z8?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Suites arithm\u00e9tiques Une suite arithm\u00e9tique de raison \\(r\\) est d\u00e9finie par : Repr\u00e9sentation graphique : les points sont align\u00e9s.Ici, la repr\u00e9sentation de la suite de premier terme 2 et de raison 0,5 ($u_n=2+0,5n$) Variations Si $r$ est positif, la suite est croissante Si $r$ est n\u00e9gatif, la suite est d\u00e9croissante Somme des termes $1+2+\\dots+n=\\dfrac{n(n+1)}{2}$ Et [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":3972,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"folder":[],"class_list":["post-4048","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/4048","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/comments?post=4048"}],"version-history":[{"count":3,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/4048\/revisions"}],"predecessor-version":[{"id":4054,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/4048\/revisions\/4054"}],"up":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/3972"}],"wp:attachment":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/media?parent=4048"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/folder?post=4048"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}