{"id":2348,"date":"2020-05-03T16:54:02","date_gmt":"2020-05-03T14:54:02","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?page_id=2348"},"modified":"2023-04-16T16:20:21","modified_gmt":"2023-04-16T14:20:21","slug":"python-calcul-des-premiers-termes-dune-suite","status":"publish","type":"page","link":"https:\/\/www.mathweb.fr\/euclide\/python-calcul-des-premiers-termes-dune-suite\/","title":{"rendered":"Python : calcul des premiers termes d’une suite"},"content":{"rendered":"\n

Nous allons voir sur cette page comment calculer \u00e0 l’aide de Python les premiers termes d’une suite, \u00e0 travers deux exemples.<\/p>\n\n\n\n

\"Calcul
Calcul des premiers termes d’une suite<\/figcaption><\/figure><\/div>\n\n\n\n\n\n\n\n

Exemple 1 : suite d\u00e9finie de fa\u00e7on explicite avec Python<\/h2>\n\n\n\n

On consid\u00e8re la suite \\((u_n)\\) d\u00e9finie pour tout entier naturel n<\/em> par : $$u_n=\\frac{n^2+3n+5}{n^2+1}.$$ Nous souhaitons calculer les 50 premiers termes de cette suite. Nous utilisons alors le programme Python suivant:<\/p>\n\n\n\n

def u(n):\n    return (n*n + 3*n + 5) \/ (n*n + 1)\n\nfor n in range(50):\n    print(\"u({}) = {}\".format(n,u(n)))<\/pre>\n\n\n\n
S’affiche alors :<\/p>\n\n\n\n
u(0) = 5.0
u(1) = 4.5
u(2) = 3.0
u(3) = 2.3
u(4) = 1.9411764705882353
u(5) = 1.7307692307692308
u(6) = 1.5945945945945945
u(7) = 1.5
u(8) = 1.4307692307692308
u(9) = 1.3780487804878048
u(10) = 1.3366336633663367
u(11) = 1.3032786885245902
u(12) = 1.2758620689655173
u(13) = 1.2529411764705882
u(14) = 1.233502538071066
u(15) = 1.2168141592920354
u(16) = 1.2023346303501945
u(17) = 1.1896551724137931
u(18) = 1.1784615384615384
u(19) = 1.1685082872928176
u(20) = 1.1596009975062345
u(21) = 1.1515837104072397
u(22) = 1.1443298969072164
u(23) = 1.1377358490566039
u(24) = 1.1317157712305026
u(25) = 1.1261980830670926
u(26) = 1.121122599704579
u(27) = 1.1164383561643836
u(28) = 1.1121019108280255
u(29) = 1.1080760095011877
u(30) = 1.1043285238623752
u(31) = 1.1008316008316008
u(32) = 1.0975609756097562
u(33) = 1.0944954128440367
u(34) = 1.0916162489196197
u(35) = 1.0889070146818924
u(36) = 1.0863531225905936
u(37) = 1.083941605839416
u(38) = 1.0816608996539792
u(39) = 1.0795006570302235
u(40) = 1.0774515927545285
u(41) = 1.0755053507728893
u(42) = 1.0736543909348442
u(43) = 1.0718918918918918
u(44) = 1.0702116675271038
u(45) = 1.0686080947680159
u(46) = 1.0670760510155881
u(47) = 1.0656108597285068
u(48) = 1.0642082429501085
u(49) = 1.062864279766861<\/pre>\n\n\n\n
J’ai ici utilis\u00e9 une fonction de formatage d’affichage pour plus de simplicit\u00e9; quand on tape en python:<\/p>\n\n\n\n
print(\"{} blabla {}\".format(var1,var2))<\/pre>\n\n\n\n
les valeurs des variables var1<\/em> et var2<\/em> s’affichent \u00e0 la place des “{}”.<\/p>\n\n\n\n
Exemple 2 : suite d\u00e9finie par r\u00e9currence (la suite de H\u00e9ron)<\/h2>\n\n\n\n
Soit \\((v_n)\\) la suite d\u00e9finie par son premier terme \\(u_0=7\\) et par la relation de r\u00e9currence:$$u_{n+1}=\\frac{1}{2}\\left(u_n+\\frac{5}{u_n}\\right)$$pour tout entier naturel n<\/em>. Nous souhaitons calculer les 10 premiers termes de cette suite; on utilise alors le programme suivant:<\/p>\n\n\n\n
# on commence par initialiser les variables\nv = 7\nprint(\"v({}) = {}\".format(0,v))\n\n# ici commence le programme de seuil\nfor n in range(1,10):\n    v = 0.5 * (v + 5 \/ v) # on utilise la relation de r\u00e9currence pour calculer le terme suivant\n    print(\"v({}) = {}\".format(n,v))<\/pre>\n\n\n\n
S’affiche alors:<\/p>\n\n\n\n
v(0) = 7
v(1) = 3.857142857142857
v(2) = 2.5767195767195767
v(3) = 2.258585661049727
v(4) = 2.2361802260811032
v(5) = 2.2360679803170367
v(6) = 2.23606797749979
v(7) = 2.23606797749979
v(8) = 2.23606797749979
v(9) = 2.23606797749979<\/pre>\n\n\n\n
\u00c0 savoir : quand on sait combien de termes on doit calculer, on utilise une boucle it\u00e9rative<\/em> “for”.<\/p>\n\n\n\n
Retrouvez sur cette page<\/a> d’autres programmes concernant ces suites : algorithmes de seuil.<\/p>\n\n\n\n
Exemple 3: la suite de Fibonacci<\/h2>\n\n\n\n
La suite de Fibonacci est ainsi d\u00e9finie pour tout entier naturel n <\/em>:$$F_0=0,\\ F_1=1,\\ F_{n+2}=F_{n+1}+F_n.$$<\/p>\n\n\n\n
On voir alors ici qu’on ne fait plus uniquement appel au terme pr\u00e9c\u00e9dent mais aussi au p\u00e9nulti\u00e8me<\/a>. Et ceci va poser quelques probl\u00e8mes. Il faut en effet extrapoler la philosophie utilis\u00e9e dans les programmes comme celui qui pr\u00e9c\u00e8de (sur la suite de H\u00e9ron). Voici une proposition:<\/p>\n\n\n\n
def fibonacci(n):\n    if n == 0:\n        return 0\n    elif n == 1:\n        return 1\n    else:\n        a, b = 0, 1\n        for k in range(2,n+1):\n            c = a + b\n            a, b = b, c\n            \n        return c<\/pre>\n\n\n\n
Il y a dans cette fonction une subtilit\u00e9 : nous avons d\u00fb introduire une variable temporaire (c<\/em>). Regardons \u00e9tape par \u00e9tape ce qui se passe pour n<\/em>=6:<\/p>\n\n\n\n
k<\/em><\/td>–<\/td>2<\/td>3<\/td>4<\/td>5<\/td>6<\/td><\/tr>
c<\/em><\/td>–<\/td>0 + 1 = 1<\/td>1 + 1 = 2<\/td>1 + 2 = 3<\/td>2 + 3 = 5<\/td>3 + 5 = 8<\/td><\/tr>
a<\/em><\/td>0<\/td>1<\/td>1<\/td>2<\/td>3<\/td>5<\/td><\/tr>
b<\/em><\/td>1<\/td>1<\/td>2<\/td>3<\/td>5<\/td>8<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
Ce programme est loin d’\u00eatre optimal car il n\u00e9cessite \u00e9norm\u00e9ment d’op\u00e9rations (surtout pour les grandes valeurs de n<\/em>), mais ce n’est pas grave. L”important est de comprendre son fonctionnement (pour les \u00e9l\u00e8ves de NSI, il existe des programmes optimis\u00e9s qui permettent de r\u00e9duire la complexit\u00e9, notamment en passant par une liste et en stockant tous les termes dans cette liste).<\/p>\n\n\n\n
N’oubliez pas que si vous souhaitez un cours de maths, vous pouvez r\u00e9server un cours sur cette page<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"
Nous allons voir sur cette page comment calculer \u00e0 l’aide de Python les premiers termes d’une suite, \u00e0 travers deux exemples.<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2348","page","type-page","status-publish","hentry"],"yoast_head":"\nPython : calcul des premiers termes d'une suite - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Comment calculer les premiers termes d'une suite \u00e0 l'aide de Python ? Avec une boucle it\u00e9rative bien s\u00fbr ! 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