Quiz — M-EASuites définies par récurrence, III -REE-JMF-13.pdf

Question 1 sur 10 20:00

[{"id":10433,"question":"Soit une suite (u_n) définie par u_0 = 2 et u_{n+1} = 3u_n - 1. Que vaut u_2 ?","option_a":"5","option_b":"14","option_c":"17","option_d":"20","option_e":"","option_f":"","bonne_reponse":"c","explication":"Calculons u_1 = 3*2 - 1 = 5, puis u_2 = 3*5 - 1 = 14. La bonne réponse est donc 14.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"5\", \"b\": \"14\", \"c\": \"17\", \"d\": \"20\"}}","_debug_options_count":4},{"id":10434,"question":"Une suite définie par récurrence est toujours monotone.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une suite définie par récurrence peut être croissante, décroissante, ou ni l'une ni l'autre (oscillante).","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":10435,"question":"La suite définie par u_{n+1} = u_n + 5 est une suite :","option_a":"géométrique","option_b":"arithmétique","option_c":"ni l'une ni l'autre","option_d":"constante","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite a une différence constante de 5 entre deux termes consécutifs, donc elle est arithmétique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"géométrique\", \"b\": \"arithmétique\", \"c\": \"ni l'une ni l'autre\",","_debug_options_count":4},{"id":10436,"question":"Soit (u_n) une suite définie par u_0 = 1 et u_{n+1} = 2u_n + 3. Que vaut u_1 ?","option_a":"5","option_b":"6","option_c":"7","option_d":"8","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons u_1 = 2*1 + 3 = 5. La bonne réponse est donc 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"5\", \"b\": \"6\", \"c\": \"7\", \"d\": \"8\"}}","_debug_options_count":4},{"id":10437,"question":"Toute suite définie par récurrence converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Certaines suites définies par récurrence divergent (ex : u_{n+1} = 2u_n avec u_0 = 1).","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":10438,"question":"La relation u_{n+1} = u_n^2 définit une suite :","option_a":"arithmétique","option_b":"géométrique","option_c":"quadratique","option_d":"linéaire","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette relation fait intervenir u_n au carré, donc la suite est dite quadratique.","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\": \"quadratique\", \"d\": \"l","_debug_options_count":4},{"id":10439,"question":"Soit (u_n) une suite définie par u_0 = 0 et u_{n+1} = u_n + n. Que vaut u_3 ?","option_a":"0","option_b":"3","option_c":"6","option_d":"9","option_e":"","option_f":"","bonne_reponse":"c","explication":"Calculons u_1 = 0 + 0 = 0, u_2 = 0 + 1 = 1, u_3 = 1 + 2 = 3. La bonne réponse est donc 3.","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\": \"3\", \"c\": \"6\", \"d\": \"9\"}}","_debug_options_count":4},{"id":10440,"question":"Une suite définie par récurrence peut être définie à partir de deux termes précédents.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Certaines suites, comme la suite de Fibonacci, sont définies par une relation de récurrence impliquant deux termes précédents.","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":10441,"question":"La suite définie par u_{n+1} = -u_n est :","option_a":"constante","option_b":"alternée","option_c":"croissante","option_d":"décroissante","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite alterne entre des valeurs positives et négatives, donc elle est alternée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"constante\", \"b\": \"alternée\", \"c\": \"croissante\", \"d\": \"décroissa","_debug_options_count":4},{"id":10442,"question":"Soit (u_n) une suite définie par u_0 = 1 et u_{n+1} = 0.5u_n + 1. Que vaut la limite de u_n quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite converge vers 2, car la limite L vérifie L = 0.5L + 1, soit L = 2.","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\": \"2\", \"d\": \"La suite diverge\"}}","_debug_options_count":4}]