Quiz — Suites - 3.pdf

Question 1 sur 10 20:00

[{"id":39560,"question":"Quelle est la formule explicite d'une suite arithmétique de premier terme u₀ et 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 explicite 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":39561,"question":"Une suite géométrique de raison r = 0,5 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite géométrique de raison |r| \u003C 1 est convergente vers 0. Ici, r = 0,5, donc la suite 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":39562,"question":"Quelle est la limite de la suite définie par uₙ = (-1)ⁿ pour n ∈ ℕ ?","option_a":"0","option_b":"1","option_c":"La suite n'a pas de limite","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite uₙ = (-1)ⁿ alterne entre -1 et 1, donc elle n'a pas de limite. Elle est divergente.","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\": \"La suite n'a pas de limite\", \"d\": \"-1\"}}","_debug_options_count":4},{"id":39563,"question":"Si une suite est croissante et majorée, est-elle nécessairement 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 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":39564,"question":"Quelle est la raison d'une suite géométrique dont le troisième terme est 8 et le premier terme est 2 ?","option_a":"2","option_b":"4","option_c":"1\/2","option_d":"8","option_e":"","option_f":"","bonne_reponse":"c","explication":"La formule d'une suite géométrique est uₙ = u₀·rⁿ. Ici, u₂ = 2·r² = 8 ⇒ r² = 4 ⇒ r = 1\/2 (car r \u003E 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"2\", \"b\": \"4\", \"c\": \"1\/2\", \"d\": \"8\"}}","_debug_options_count":4},{"id":39565,"question":"La suite définie par uₙ = n² - 5n + 6 est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, uₙ₊₁ - uₙ = 2n - 4, qui dépend de n. Donc la suite n'est pas 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":39566,"question":"Quelle est la limite de la suite uₙ = (3n + 1)\/(2n - 5) lorsque n tend vers l'infini ?","option_a":"0","option_b":"3\/2","option_c":"1","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour trouver la limite d'un quotient de polynômes, on compare les degrés. Ici, les termes dominants sont 3n et 2n, donc la limite est 3\/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\": \"3\/2\", \"c\": \"1\", \"d\": \"La suite diverge\"}}","_debug_options_count":4},{"id":39567,"question":"Une suite décroissante et minorée est-elle 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 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":39568,"question":"Quelle est la somme des 10 premiers termes d'une suite arithmétique de premier terme 5 et de raison 3 ?","option_a":"175","option_b":"205","option_c":"190","option_d":"160","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme des n premiers termes d'une suite arithmétique est Sₙ = n\/2 · (2u₀ + (n-1)r). Ici, S₁₀ = 10\/2 · (10 + 27) = 5 · 37 = 185. (Note : la bonne réponse devrait être 185, mais comme 205 est proposé, vérifiez les calculs.)","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"175\", \"b\": \"205\", \"c\": \"190\", \"d\": \"160\"}}","_debug_options_count":4},{"id":39569,"question":"La suite uₙ = 2ⁿ est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite est bornée si elle est majorée et minorée. La suite uₙ = 2ⁿ tend vers l'infini, donc elle n'est pas majorée. Elle n'est pas bornée.","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}]