{"id":9640,"date":"2024-08-15T19:03:56","date_gmt":"2024-08-15T18:03:56","guid":{"rendered":"https:\/\/mathsguyon.fr\/?page_id=9640"},"modified":"2026-09-05T16:03:41","modified_gmt":"2026-09-05T15:03:41","slug":"raisonnement-par-recurrence-et-suites","status":"publish","type":"page","link":"https:\/\/mathsguyon.fr\/?page_id=9640","title":{"rendered":"Suites et r\u00e9currence"},"content":{"rendered":"\n<h1 class=\"wp-block-heading has-text-align-center\"><a href=\"https:\/\/www.dropbox.com\/scl\/fi\/i6c59nrylgw3cq1y6hc6e\/01_Recurrence_mathsguyon_2026_relu2.pdf?rlkey=9qwowot6zm6f8cf3lhz5gydhp&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/i6c59nrylgw3cq1y6hc6e\/01_Recurrence_mathsguyon_2026_relu2.pdf?rlkey=9qwowot6zm6f8cf3lhz5gydhp&amp;dl=0\">Le cours<\/a>         <a href=\"https:\/\/www.dropbox.com\/scl\/fi\/htbx2e8tgn50zf71ugmx3\/PDT_2026_new.pdf?rlkey=g9xk6autno0dohenszqrxokr5&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/htbx2e8tgn50zf71ugmx3\/PDT_2026_new.pdf?rlkey=g9xk6autno0dohenszqrxokr5&amp;dl=0\">Le Plan de travail<\/a><\/h1>\n\n\n\n<iframe loading=\"lazy\" src=\"https:\/\/coopmaths.fr\/alea\/?uuid=ed3cf&amp;id=TSA1-12&amp;alea=dUAO&amp;uuid=4c1f5&amp;id=TSA1-13&amp;alea=UxCc&amp;uuid=a73e2&amp;id=TSA1-14&amp;alea=97NI&amp;uuid=787b5&amp;id=TSA1-11&amp;alea=fgH5&amp;uuid=7f24e&amp;id=TSA1-10&amp;alea=HR4G&amp;v=eleve&amp;es=101100110&amp;title=Synth%25C3%25A8se%2520raisonnement%2520par%2520r%25C3%25A9currence%2520MathAL%25C3%2589A\" width=\"600\" height=\"600\" frameborder=\"0\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exercice <\/strong>1 : <a href=\"https:\/\/www.dropbox.com\/scl\/fi\/hv2dft83l5zmq1663sv20\/eval_formative_exo3.pdf?rlkey=iu6aadsheas7bkowennmbfnac&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/hv2dft83l5zmq1663sv20\/eval_formative_exo3.pdf?rlkey=iu6aadsheas7bkowennmbfnac&amp;dl=0\">Correction<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D\u00e9montrer que, pour tout entier $n \\geqslant 1$ : $\\displaystyle\\sum_{k=1}^{n} \\frac{1}{(2k-1)(2k+1)}=\\dfrac{n}{2 n+1}$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exercice 2 :<\/strong> <a href=\"https:\/\/www.dropbox.com\/scl\/fi\/wq31qnkn0tbfbx6j0dusu\/eval_formative_exo4.pdf?rlkey=tt7md2f54uzmr94bcovzya99p&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/wq31qnkn0tbfbx6j0dusu\/eval_formative_exo4.pdf?rlkey=tt7md2f54uzmr94bcovzya99p&amp;dl=0\">Correction<\/a><br>$u_0=2$ et $u_{n+1}=\\dfrac{u_n}{1+u_n}$ pour tout entier naturel $n$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D\u00e9montrer par r\u00e9currence que, pour tout entier naturel $n$, $u_n=\\dfrac{2}{2 n+1}.$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exercice 3 :<\/strong> <a href=\"https:\/\/www.dropbox.com\/scl\/fi\/t9dzmct34rsxki3kw7jjc\/eval_formative_exo5.pdf?rlkey=l2fcvygmwdb6h6kzmpagvt06f&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/t9dzmct34rsxki3kw7jjc\/eval_formative_exo5.pdf?rlkey=l2fcvygmwdb6h6kzmpagvt06f&amp;dl=0\">Correction<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re la suite ( $u_n$ ) d\u00e9finie par $u_0=1$ et $u_{n+1}=\\dfrac{1}{4} u_n+3$ pour tout entier $n \\geqslant 0$.<br>Montrer par r\u00e9currence pour tout entier $n \\geq 1$, $3 \\leq u_n \\leq 4$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exercice 4 :<\/strong> <a href=\"https:\/\/www.dropbox.com\/scl\/fi\/p8r6evn7z957v8g01v9y0\/eval_formative_exo6.pdf?rlkey=43882wds44jiuoq6tp6rbnl9p&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/t9dzmct34rsxki3kw7jjc\/eval_formative_exo5.pdf?rlkey=l2fcvygmwdb6h6kzmpagvt06f&amp;dl=0\">Correction<\/a><br>On consid\u00e8re la suite ( $S_n$ ) d\u00e9finie pour tout entier $n \\geq 1$ par $S_n=\\displaystyle\\sum_{k=1}^n(2 k-1)$<br>D\u00e9montrer par r\u00e9currence que, pour tout entier $n \\geq 1$, on a $ S_n=n^2$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exercice 5 :<\/strong> <a href=\"https:\/\/www.dropbox.com\/scl\/fi\/4cg1o79tpywc9omx5j258\/eval_formative_exo7.pdf?rlkey=pbspymjs96h9dz9ohi5pim7hf&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/4cg1o79tpywc9omx5j258\/eval_formative_exo7.pdf?rlkey=pbspymjs96h9dz9ohi5pim7hf&amp;dl=0\">Correction<\/a><br>Soit $\\left(u_n\\right)$ la suite d\u00e9finie par $u_0=2$ et pour tout $n$ de $\\mathbb{N}$, $u_{n+1}=\\dfrac{1}{3} u_n+2$<br>D\u00e9montrer par r\u00e9currence que la suite $\\left(u_n\\right)$ est croissante.<\/p>\n\n\n<h2 style=\"text-align: center;\"><strong data-rich-text-format-boundary=\"true\"><span style=\"color: #800000;\">Correction exercices suites arithm\u00e9tico-g\u00e9om\u00e9triques<\/span><br \/><\/strong><\/h2>\n\n\n<p class=\"wp-block-paragraph\"><strong>Exercice 1 : <\/strong> <a href=\"https:\/\/www.dropbox.com\/scl\/fi\/ci9q31dd2xbti0o2diwwp\/03_SAG_2024_exo_corrige.pdf?rlkey=0hshqq8fhz2otp9qi2lyxrng1&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/ci9q31dd2xbti0o2diwwp\/03_SAG_2024_exo_corrige.pdf?rlkey=0hshqq8fhz2otp9qi2lyxrng1&amp;dl=0\">Correction<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re une suite $\\left(u_n\\right)$ d\u00e9finie sur $\\mathbb{N}$ par : <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$u_0=4$ et $u_{n+1}=2u_n-3$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soit la suite $\\left(v_n\\right)$ d\u00e9finie sur N par: $v_n=u_n-3$.<br>1. Quelle est la nature de la suite $\\left(u_n\\right)$.<br>2. Montrer que la suite $\\left(v_n\\right)$ est g\u00e9om\u00e9trique.<br>3. Donner l&rsquo;expression de $v_n$ en fonction de $n$.<br>4. En d\u00e9duire l&rsquo;expression de $u_n$ en fonction de $n$.<br>5. Calculer la somme des 11 premiers termes de $\\left(u_n\\right)$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Exercice 2:  <a href=\"https:\/\/www.dropbox.com\/scl\/fi\/cjsz0k3bgb5en877ya8gp\/03_SAG_2024_exo_corrige2.pdf?rlkey=7xf2kjurq6g241dxcxn07dtl6&amp;dl=0\" data-type=\"link\" data-id=\"https:\/\/www.dropbox.com\/scl\/fi\/cjsz0k3bgb5en877ya8gp\/03_SAG_2024_exo_corrige2.pdf?rlkey=7xf2kjurq6g241dxcxn07dtl6&amp;dl=0\">Correction<\/a><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\left(u_n\\right)$ est la suite d\u00e9finie sur $\\mathbb{N}$ par $u_0=5$ et pour tout entier naturel $n$, $$u_{n+1}=\\frac{1}{2} u_n+4$$<br>1. Calculer $u_1, u_2, u_3$ et $u_4$.<br>2. On pose, pour tout $n \\in \\mathbb{N}$, $v_n=u_n-8$.<br>a. Montrer que la suite $\\left(v_n\\right)$ est une suite g\u00e9om\u00e9trique de raison $\\frac{1}{2}$.<br>b. Exprimer $v_n$ en fonction de $n$.<br>3. a. Exprimer $u_n$ en fonction de n.<br>b. Calculer $u_{10}$<br>4. D\u00e9terminer les variations de la suite $\\left(u_n\\right)$<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Le cours Le Plan de travail Exercice 1 : Correction D\u00e9montrer que, pour tout entier $n \\geqslant 1$ : $\\displaystyle\\sum_{k=1}^{n} \\frac{1}{(2k-1)(2k+1)}=\\dfrac{n}{2 n+1}$ Exercice 2 : Correction$u_0=2$ et $u_{n+1}=\\dfrac{u_n}{1+u_n}$ pour tout entier naturel $n$ D\u00e9montrer par r\u00e9currence que, pour tout entier &hellip; <a href=\"https:\/\/mathsguyon.fr\/?page_id=9640\">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-9640","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/9640","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=9640"}],"version-history":[{"count":80,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/9640\/revisions"}],"predecessor-version":[{"id":10717,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/9640\/revisions\/10717"}],"wp:attachment":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9640"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}