Quiz — 1569925209_8-monotonie-dune-suite-methode-3.pdf

Question 1 sur 10 20:00

[{"id":92202,"question":"Quelle est la condition pour qu'une suite (u_n) soit croissante ?","option_a":"u_{n+1} \u003E u_n pour tout n","option_b":"u_{n+1} ≥ u_n pour tout n","option_c":"u_{n+1} ≤ u_n pour tout n","option_d":"u_{n+1} = u_n pour tout n","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite est croissante si chaque terme est supérieur ou égal au précédent. La condition u_{n+1} ≥ u_n pour tout n est donc la bonne réponse.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"u_{n+1} \u003E u_n pour tout n\", \"b\": \"u_{n+1} ≥ u_n pour tout n\", \"","_debug_options_count":4},{"id":92203,"question":"La suite définie par u_n = n² est-elle croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour tout n, u_{n+1} = (n+1)² = n² + 2n + 1 \u003E n² = u_n. La suite est donc strictement croissante.","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":92204,"question":"Soit la suite u_n = 2n + 1. Quelle méthode permet d'étudier sa monotonie ?","option_a":"Étudier le signe de u_{n+1} - u_n","option_b":"Calculer la dérivée de u_n","option_c":"Comparer u_n à une suite arithmétique","option_d":"Étudier les limites de u_n","option_e":"","option_f":"","bonne_reponse":"a","explication":"La méthode la plus simple consiste à calculer u_{n+1} - u_n = 2(n+1) + 1 - (2n + 1) = 2 \u003E 0. La suite est donc croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Étudier le signe de u_{n+1} - u_n\", \"b\": \"Calculer la dérivée ","_debug_options_count":4},{"id":92205,"question":"La suite v_n = (-1)^n est-elle monotone ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite alterne entre -1 et 1, elle n'est ni croissante ni décroissante. Elle n'est donc pas monotone.","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":92206,"question":"Pour étudier la monotonie d'une suite définie par u_{n+1} = f(u_n), quelle méthode est adaptée ?","option_a":"Étudier la fonction f","option_b":"Calculer u_{n+1} - u_n","option_c":"Comparer u_n à une suite géométrique","option_d":"Étudier les limites de f","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il faut étudier la fonction f pour comprendre comment u_{n+1} évolue par rapport à u_n. Si f est croissante, la monotonie de la suite dépend de u_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\": \"Étudier la fonction f\", \"b\": \"Calculer u_{n+1} - u_n\", \"c\": \"Com","_debug_options_count":4},{"id":92207,"question":"Soit la suite w_n = 1\/n. Quelle est sa monotonie ?","option_a":"Croissante","option_b":"Décroissante","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour tout n, w_{n+1} = 1\/(n+1) \u003C 1\/n = w_n. La suite est donc strictement décroissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Croissante\", \"b\": \"Décroissante\", \"c\": \"Constante\", \"d\": \"Non mo","_debug_options_count":4},{"id":92208,"question":"La suite u_n = n\/(n+1) est-elle monotone ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"On peut montrer que u_{n+1} - u_n = 1\/((n+1)(n+2)) \u003E 0 pour tout n. La suite est donc strictement croissante.","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":92209,"question":"Pour comparer deux suites u_n et v_n, quelle méthode est la plus efficace ?","option_a":"Étudier u_n - v_n","option_b":"Calculer u_n \/ v_n","option_c":"Étudier les limites de u_n et v_n","option_d":"Comparer leurs termes initiaux","option_e":"","option_f":"","bonne_reponse":"a","explication":"Étudier u_n - v_n permet de déterminer laquelle des deux suites est la plus grande à partir d'un certain rang.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Étudier u_n - v_n\", \"b\": \"Calculer u_n \/ v_n\", \"c\": \"Étudier le","_debug_options_count":4},{"id":92210,"question":"Soit la suite u_n = (-2)^n. Quelle est sa monotonie ?","option_a":"Croissante","option_b":"Décroissante","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"d","explication":"La suite alterne entre des valeurs positives et négatives de plus en plus grandes en valeur absolue. Elle n'est donc pas monotone.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"Croissante\", \"b\": \"Décroissante\", \"c\": \"Constante\", \"d\": \"Non mo","_debug_options_count":4},{"id":92211,"question":"La suite u_n = n² - 5n + 6 est-elle monotone pour n ≥ 3 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour n ≥ 3, u_{n+1} - u_n = 2n - 4 ≥ 2 \u003E 0. La suite est donc strictement croissante à partir du rang 3.","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}]