Quiz — 4. Corrections suites 2.pdf

Question 1 sur 10 20:00

[{"id":87133,"question":"Soit une suite (uₙ) définie par uₙ = 3n + 2. Quelle est la nature de cette suite ?","option_a":"Arithmétique de raison 3","option_b":"Géométrique de raison 3","option_c":"Arithmétique de raison 2","option_d":"Ni arithmétique ni géométrique","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite uₙ = 3n + 2 est de la forme uₙ = an + b, donc elle est arithmétique de raison a = 3. Le terme constant b = 2 n’influence pas la nature de la suite.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Arithmétique de raison 3\", \"b\": \"Géométrique de raison 3\", \"c\"","_debug_options_count":4},{"id":87134,"question":"La suite (vₙ) définie par vₙ = (-1)ⁿ est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite (vₙ) alterne entre -1 et 1, donc elle n’a pas de limite finie. 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":87135,"question":"Quelle est la limite de la suite (wₙ) définie par wₙ = (2n² + 3n) \/ (n² - 5) quand n tend vers l’infini ?","option_a":"0","option_b":"2","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant le numérateur et le dénominateur par n², on obtient wₙ = (2 + 3\/n) \/ (1 - 5\/n²). Quand n → ∞, 3\/n et 5\/n² tendent vers 0, donc la limite est 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\": \"2\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":87136,"question":"Soit une suite géométrique (uₙ) de raison q = 1\/2 et de premier terme u₀ = 8. Quelle est la valeur de u₃ ?","option_a":"1","option_b":"2","option_c":"4","option_d":"8","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une suite géométrique, uₙ = u₀ × qⁿ. Donc u₃ = 8 × (1\/2)³ = 8 × 1\/8 = 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\": \"1\", \"b\": \"2\", \"c\": \"4\", \"d\": \"8\"}}","_debug_options_count":4},{"id":87137,"question":"Toute suite croissante est 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 +∞. Par exemple, la suite uₙ = n est croissante et 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":87138,"question":"Quelle est la somme des 10 premiers termes d’une suite arithmétique de premier terme u₀ = 5 et de raison r = 3 ?","option_a":"190","option_b":"205","option_c":"220","option_d":"235","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.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"190\", \"b\": \"205\", \"c\": \"220\", \"d\": \"235\"}}","_debug_options_count":4},{"id":87139,"question":"Soit une suite (uₙ) définie par uₙ₊₁ = 2uₙ + 1 avec u₀ = 1. Quelle est la valeur de u₂ ?","option_a":"5","option_b":"7","option_c":"9","option_d":"11","option_e":"","option_f":"","bonne_reponse":"b","explication":"u₁ = 2×1 + 1 = 3, puis u₂ = 2×3 + 1 = 7. La suite est définie par récurrence.","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\": \"7\", \"c\": \"9\", \"d\": \"11\"}}","_debug_options_count":4},{"id":87140,"question":"Si une suite (uₙ) est bornée et monotone, 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 limite monotone : une suite bornée et monotone est convergente. La réponse correcte est donc 'Vrai'.","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":87141,"question":"Quelle est la limite de la suite (uₙ) définie par uₙ = (3ⁿ + 2ⁿ) \/ 4ⁿ quand n tend vers l’infini ?","option_a":"0","option_b":"1\/2","option_c":"3\/4","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"En factorisant par 4ⁿ, on obtient uₙ = ( (3\/4)ⁿ + (2\/4)ⁿ ). Comme 3\/4 \u003C 1 et 2\/4 \u003C 1, les deux termes tendent vers 0 quand n → ∞.","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\/2\", \"c\": \"3\/4\", \"d\": \"1\"}}","_debug_options_count":4},{"id":87142,"question":"Soit une suite (vₙ) définie par vₙ = n² - 5n + 6. Pour quelles valeurs de n cette suite est-elle positive ?","option_a":"n ≤ 2 ou n ≥ 3","option_b":"n = 2 ou n = 3","option_c":"n \u003C 2 ou n \u003E 3","option_d":"Toujours positive","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite vₙ = n² - 5n + 6 est un polynôme du second degré. Ses racines sont n = 2 et n = 3. Elle est positive à l’extérieur des racines, soit n ≤ 2 ou n ≥ 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\": \"n ≤ 2 ou n ≥ 3\", \"b\": \"n = 2 ou n = 3\", \"c\": \"n \u003C 2 ou n \u003E 3\"","_debug_options_count":4}]