Quiz — Suites -1.pdf

Question 1 sur 10 20:00

[{"id":43005,"question":"Quelle est la limite de la suite définie par u_n = (2n + 1)\/(n + 3) lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite est obtenue en divisant le numérateur et le dénominateur par n, ce qui donne (2 + 1\/n)\/(1 + 3\/n) → 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":43006,"question":"Une suite croissante est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite croissante peut diverger vers +∞ (exemple : u_n = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":43007,"question":"Soit une suite géométrique de raison r = -0.5. Que peut-on dire de sa convergence ?","option_a":"Elle converge vers 0","option_b":"Elle diverge vers +∞","option_c":"Elle est constante","option_d":"Elle n'a pas de limite","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si |r| \u003C 1, la suite géométrique converge vers 0, quelle que soit la valeur initiale.","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 vers +∞\", \"c\": \"Elle ","_debug_options_count":4},{"id":43008,"question":"Quelle est la formule de la somme des n premiers termes d'une suite arithmétique de premier terme u_0 et de raison r ?","option_a":"S_n = n(u_0 + u_n)\/2","option_b":"S_n = n(u_0 + r)\/2","option_c":"S_n = (u_0 + u_n)\/2","option_d":"S_n = n²(u_0 + r)\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule correcte est S_n = n(u_0 + u_n)\/2, où u_n = u_0 + n*r.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"S_n = n(u_0 + u_n)\/2\", \"b\": \"S_n = n(u_0 + r)\/2\", \"c\": \"S_n = (u_","_debug_options_count":4},{"id":43009,"question":"Toute suite bornée est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite bornée peut ne pas avoir de limite (exemple : u_n = (-1)^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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":43010,"question":"Soit la suite définie par u_0 = 1 et u_{n+1} = 2u_n + 1. Quelle est la limite de cette suite ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"d","explication":"La suite est strictement croissante et non bornée, donc elle diverge vers +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":43011,"question":"Quelle est la nature de la suite définie par u_n = n² - 5n + 6 ?","option_a":"Arithmétique","option_b":"Géométrique","option_c":"Ni l'une ni l'autre","option_d":"Constante","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette suite est un polynôme du second degré, donc ni arithmétique ni géométrique.","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\", \"b\": \"Géométrique\", \"c\": \"Ni l'une ni l'autre\",","_debug_options_count":4},{"id":43012,"question":"Une suite décroissante et minorée est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème de la limite monotone, une suite décroissante et minorée converge.","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":43013,"question":"Soit une suite géométrique de premier terme u_0 = 3 et de raison r = 2. Quel est le 5ème terme ?","option_a":"48","option_b":"96","option_c":"12","option_d":"192","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le 5ème terme est u_4 = u_0 * r^4 = 3 * 2^4 = 3 * 16 = 48. Attention à l'indexation (n commence à 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"48\", \"b\": \"96\", \"c\": \"12\", \"d\": \"192\"}}","_debug_options_count":4},{"id":43014,"question":"Quelle est la limite de la suite définie par u_n = (3^n + 2^n)\/(5^n) lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"3\/5","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant par 5^n, on obtient (3\/5)^n + (2\/5)^n → 0 + 0 = 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\": \"1\", \"c\": \"3\/5\", \"d\": \"∞\"}}","_debug_options_count":4}]