Quiz — suite 3.pdf

Question 1 sur 10 20:00

[{"id":103084,"question":"Quelle est la formule générale d'une suite arithmétique de raison r ?","option_a":"uₙ = u₀ + n","option_b":"uₙ = u₀ + n·r","option_c":"uₙ = u₀·rⁿ","option_d":"uₙ = u₀ + rⁿ","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule d'une suite arithmétique est uₙ = u₀ + n·r, où u₀ est le premier terme et r la raison.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"uₙ = u₀ + n\", \"b\": \"uₙ = u₀ + n·r\", \"c\": \"uₙ = u₀·r","_debug_options_count":4},{"id":103085,"question":"La suite définie par uₙ = (-1)ⁿ est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite alterne entre -1 et 1, elle n'a pas de limite finie, donc elle est divergente.","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":103086,"question":"Quel est le terme général d'une suite géométrique de raison q ?","option_a":"uₙ = u₀ + qⁿ","option_b":"uₙ = u₀·qⁿ","option_c":"uₙ = u₀ + n·q","option_d":"uₙ = u₀·n·q","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule d'une suite géométrique est uₙ = u₀·qⁿ, où u₀ est le premier terme et q la raison.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"uₙ = u₀ + qⁿ\", \"b\": \"uₙ = u₀·qⁿ\", \"c\": \"uₙ = u₀ ","_debug_options_count":4},{"id":103087,"question":"Si une suite est croissante et majorée, alors elle est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de la convergence monotone : une suite croissante et majorée est convergente.","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":103088,"question":"Quelle est la limite de la suite uₙ = (3ⁿ + 1)\/(2·3ⁿ) quand n tend vers l'infini ?","option_a":"0","option_b":"1\/2","option_c":"3\/2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par 3ⁿ, on obtient (1 + 1\/3ⁿ)\/2, qui tend vers 1\/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\/2\", \"c\": \"3\/2\", \"d\": \"1\"}}","_debug_options_count":4},{"id":103089,"question":"Une suite géométrique de raison q = 1 est constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si q = 1, alors uₙ = u₀ pour tout n, donc la suite est constante (et convergente vers u₀).","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":103090,"question":"Quel théorème permet de déterminer la limite d'une suite définie par uₙ₊₁ = f(uₙ) ?","option_a":"Théorème des gendarmes","option_b":"Théorème de convergence monotone","option_c":"Théorème du point fixe","option_d":"Théorème de comparaison","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le théorème du point fixe s'applique aux suites définies par récurrence uₙ₊₁ = f(uₙ) sous certaines conditions.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Théorème des gendarmes\", \"b\": \"Théorème de convergence monoto","_debug_options_count":4},{"id":103091,"question":"La suite uₙ = n² - 5n + 6 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite est un polynôme de degré 2, elle tend vers +∞ quand n tend vers l'infini, donc elle est divergente.","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":103092,"question":"Quelle est la raison d'une suite géométrique où u₃ = 8 et u₅ = 32 ?","option_a":"2","option_b":"3","option_c":"4","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"On a u₅ = u₃·q² ⇒ 32 = 8·q² ⇒ q² = 4 ⇒ q = 2 (car la raison est positive).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"4\", \"d\": \"1\/2\"}}","_debug_options_count":4},{"id":103093,"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 : uₙ = (-1)ⁿ est bornée mais divergente).","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}]