{"id":7373,"date":"2020-05-08T19:32:41","date_gmt":"2020-05-08T18:32:41","guid":{"rendered":"http:\/\/mathsguyon.fr\/?page_id=7373"},"modified":"2020-05-09T11:55:26","modified_gmt":"2020-05-09T10:55:26","slug":"essai-suites","status":"publish","type":"page","link":"https:\/\/mathsguyon.fr\/?page_id=7373","title":{"rendered":"Somme des termes d&rsquo;une suite arithm\u00e9tique"},"content":{"rendered":"<h2><strong>Somme de termes cons\u00e9cutifs :<\/strong><\/h2>\n<h2><strong><span style=\"color: #800000;\">Cas particulier :<\/span><\/strong><\/h2>\n<p>Pour tout entier naturel n on a : \\[1+2+\\cdots + n = \\frac{n \\left( n +1\\right)}{2}\\]<\/p>\n<h2><span style=\"color: #800000;\"><strong>Cas g\u00e9n\u00e9ral<\/strong><\/span><\/h2>\n<p>Soit \\(\\left( u_n \\right)\\) une suite arithm\u00e9tique de premier terme \\(u_0\\), \\[ u_0 + u_{1} + \\cdots + u_n =\\left( n +1\\right) \\times \\frac{ u_0 +u_n }{2}\\]<br \/>\nLa somme de termes cons\u00e9cutifs d&rsquo;une suite arithm\u00e9tique est \u00e9gale \u00e0 :\\[\\text{nombre de termes }\\:\\times \\frac{\\text{premier terme}\\:+\\:\\text{dernier terme}}{2}\\]<\/p>\n<p><strong>Preuve :<\/strong><\/p>\n<p>Soit \\(\\left(u_n \\right)\\) une suite arithm\u00e9tique de premier terme \\(u_0\\) et de raison \\(r\\).<br \/>\n\\[ \\begin{split}<br \/>\nS &amp;= u_{0} + u_1 +u_2 +\\cdots +u_n\\\\<br \/>\n&amp;= u_{0} + \\left( u_0+r\\right)+ \\left( u_0+2r\\right) +\\cdots +\\left(u_0+nr\\right) \\\\<br \/>\n&amp;= (n+1) u_0 + \\left( 1+ 2 +\\cdots +n\\right)\\times r \\\\<br \/>\n&amp; =(n+1) u_0 +\\frac{n \\left( n +1\\right)}{2}\\times r \\\\<br \/>\n&amp; =(n+1) \\left(\\frac{2 u_0+ nr}{2}\\right) \\\\<br \/>\n&amp; =(n+1) \\left(\\frac{u_0+ u_n}{2}\\right) \\\\<br \/>\n\\end{split}<br \/>\n\\]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Somme de termes cons\u00e9cutifs : Cas particulier : Pour tout entier naturel n on a : \\[1+2+\\cdots + n = \\frac{n \\left( n +1\\right)}{2}\\] Cas g\u00e9n\u00e9ral Soit \\(\\left( u_n \\right)\\) une suite arithm\u00e9tique de premier terme \\(u_0\\), \\[ u_0 + &hellip; <a href=\"https:\/\/mathsguyon.fr\/?page_id=7373\">Continuer la lecture <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-7373","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/7373","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7373"}],"version-history":[{"count":23,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/7373\/revisions"}],"predecessor-version":[{"id":7466,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/7373\/revisions\/7466"}],"wp:attachment":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7373"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}