Quiz — Série N°5_(Suites).pdf

Question 1 sur 10 20:00

[{"id":58315,"question":"Soit une suite (u_n) définie par u_0 = 2 et u_{n+1} = 3u_n - 1. Quelle est la nature de cette suite ?","option_a":"Arithmétique de raison 3","option_b":"Géométrique de raison 3","option_c":"Ni arithmétique ni géométrique","option_d":"Arithmétique de raison -1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite n'est ni arithmétique (car u_{n+1} - u_n ≠ constante) ni géométrique (car u_{n+1}\/u_n ≠ constante). Elle est définie par une relation de récurrence linéaire du premier ordre.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Arithmétique de raison 3\", \"b\": \"Géométrique de raison 3\", \"c\"","_debug_options_count":4},{"id":58316,"question":"Une suite croissante et majorée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental : toute suite croissante et majorée converge vers sa borne supérieure.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":58317,"question":"Soit (u_n) une suite géométrique de raison q = 0.5 et u_0 = 4. Quelle est la limite de (u_n) quand n tend vers l'infini ?","option_a":"0","option_b":"4","option_c":"2","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"a","explication":"Puisque |q| = 0.5 \u003C 1, la suite géométrique converge vers 0. La limite est donc 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"4\", \"c\": \"2\", \"d\": \"La suite diverge\"}}","_debug_options_count":4},{"id":58318,"question":"Soit (u_n) une suite définie par u_n = (-1)^n \/ n. Que peut-on dire de sa convergence ?","option_a":"Elle converge vers 0","option_b":"Elle diverge","option_c":"Elle converge vers 1","option_d":"Elle n'est pas définie","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est bornée et tend vers 0 car |u_n| = 1\/n → 0. Elle converge donc vers 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Elle converge vers 0\", \"b\": \"Elle diverge\", \"c\": \"Elle converge v","_debug_options_count":4},{"id":58319,"question":"Le théorème de comparaison s'applique uniquement aux suites croissantes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème de comparaison s'applique à toute suite, qu'elle soit croissante ou décroissante, tant que les inégalités sont respectées.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":58320,"question":"Soit (u_n) une suite définie par u_{n+1} = u_n + n et u_0 = 1. Que vaut u_3 ?","option_a":"4","option_b":"7","option_c":"10","option_d":"13","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons les termes : u_1 = 1 + 0 = 1, u_2 = 1 + 1 = 2, u_3 = 2 + 2 = 4. La réponse correcte est 4 (erreur dans les options).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"4\", \"b\": \"7\", \"c\": \"10\", \"d\": \"13\"}}","_debug_options_count":4},{"id":58321,"question":"Une suite bornée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite bornée n'est pas nécessairement convergente (exemple : la suite (-1)^n est bornée mais diverge).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":58322,"question":"Soit (u_n) une suite arithmétique de raison r = -2 et u_5 = 10. Que vaut u_0 ?","option_a":"18","option_b":"20","option_c":"22","option_d":"24","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule d'une suite arithmétique est u_n = u_0 + n*r. Donc u_5 = u_0 + 5*(-2) = 10 → u_0 = 10 + 10 = 20.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"18\", \"b\": \"20\", \"c\": \"22\", \"d\": \"24\"}}","_debug_options_count":4},{"id":58323,"question":"Soit (u_n) une suite définie par u_n = n^2 \/ (n^2 + 1). Quelle est sa limite quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"1\/2","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n^2, on obtient u_n = 1 \/ (1 + 1\/n^2) → 1 quand n → ∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"1\/2\", \"d\": \"La suite diverge\"}}","_debug_options_count":4},{"id":58324,"question":"Soit (u_n) une suite définie par u_{n+1} = (u_n + 2)\/3 et u_0 = 0. Cette suite converge-t-elle ? Si oui, vers quelle valeur ?","option_a":"Oui, vers 1","option_b":"Oui, vers 0.5","option_c":"Non, elle diverge","option_d":"Oui, vers 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite converge vers la solution de l'équation L = (L + 2)\/3, soit L = 1. C'est un point fixe attractif.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Oui, vers 1\", \"b\": \"Oui, vers 0.5\", \"c\": \"Non, elle diverge\", \"d\"","_debug_options_count":4}]