{"id":775,"date":"2019-11-02T17:55:53","date_gmt":"2019-11-02T16:55:53","guid":{"rendered":"http:\/\/caneri.gmollet.free.fr\/?page_id=775"},"modified":"2019-11-02T17:55:53","modified_gmt":"2019-11-02T16:55:53","slug":"activite-ix","status":"publish","type":"page","link":"https:\/\/courscaneri.ovh\/?page_id=775","title":{"rendered":"Activit\u00e9 IX – Les suites"},"content":{"rendered":"

On rappelle que \\displaystyle \\sum_{k=1}^nk = \\frac{n(n+1)}{2}<\/code>.<\/p>\n

I – Calcul de la somme des n<\/code> premiers carr\u00e9s de nombres entiers naturels non nuls<\/h2>\n

Soit n \\in \\mathbb{N}<\/code>. Soit k \\in [1;n]<\/code>.<\/p>\n

    \n
  1. D\u00e9velopper (k+1)^3<\/code>.<\/li>\n
  2. En d\u00e9duire l’expression de (k+1)^3-k^3<\/code> en fonction de k<\/code>.<\/li>\n
  3. D\u00e9terminer de deux fa\u00e7ons diff\u00e9rentes \\displaystyle \\sum_{k=1}^n\\left[(k+1)^3-k^3\\right]<\/code> en fonction de n<\/code>.<\/li>\n
  4. En d\u00e9duire \\displaystyle \\sum_{k=1}^nk^2<\/code>.<\/li>\n
  5. Calculer \\displaystyle \\sum_{k=1}^{20}k^2<\/code><\/li>\n<\/ol>\n

    II – Calcul de la somme des n<\/code> premiers cubes de nombres entiers naturels non nuls<\/h2>\n

    Soit n \\in \\mathbb{N}<\/code>. Soit k \\in [1;n]<\/code>.<\/p>\n

      \n
    1. D\u00e9velopper (k+1)^4<\/code>.<\/li>\n
    2. En d\u00e9duire l’expression de (k+1)^4-k^4<\/code> en fonction de k<\/code>.<\/li>\n
    3. D\u00e9terminer de deux fa\u00e7ons diff\u00e9rentes \\displaystyle \\sum_{k=1}^n\\left[(k+1)^4-k^4\\right]<\/code> en fonction de n<\/code>.<\/li>\n
    4. En d\u00e9duire \\displaystyle \\sum_{k=1}^nk^3<\/code>.<\/li>\n
    5. Calculer \\displaystyle \\sum_{k=1}^{20}k^3<\/code><\/li>\n<\/ol>\n

      III – Calcul de la somme des n<\/code> premi\u00e8res puissances quatri\u00e8me de nombres entiers naturels non nuls<\/h2>\n
        \n
      1. Trouvez une formule pour \\displaystyle \\sum_{k=1}^nk^4<\/code>.<\/li>\n
      2. Calculer \\displaystyle \\sum_{k=1}^{20}k^4<\/code>.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

        On rappelle que \\displaystyle \\sum_{k=1}^nk = \\frac{n(n+1)}{2}. I – Calcul de la somme des n premiers carr\u00e9s de nombres entiers naturels non nuls Soit n \\in \\mathbb{N}. Soit k \\in [1;n]. D\u00e9velopper (k+1)^3. En d\u00e9duire l’expression de (k+1)^3-k^3 en fonction de k. D\u00e9terminer de deux fa\u00e7ons diff\u00e9rentes \\displaystyle \\sum_{k=1}^n\\left[(k+1)^3-k^3\\right] en fonction de n. En d\u00e9duire […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":276,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-775","page","type-page","status-publish","czr-hentry"],"_links":{"self":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/775","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=775"}],"version-history":[{"count":0,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/775\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/276"}],"wp:attachment":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=775"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}