Maîtrisez les suites réelles : limites, convergence et applications

Question 1 sur 10 10:00

[{"id":76022,"question":"Soit la suite définie par uₙ = 3n + 2. Quelle est sa nature ?","option_a":"Suite arithmétique de raison 3","option_b":"Suite géométrique de raison 3","option_c":"Suite constante","option_d":"Suite divergente","option_e":"","option_f":"","bonne_reponse":"A","explication":"La suite est arithmétique car la différence entre deux termes consécutifs est constante (uₙ₊₁ - uₙ = 3).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76023,"question":"La suite uₙ = (-1)ⁿ est-elle convergente ?","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'a donc pas de limite finie et est divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76024,"question":"Quelle est la limite de la suite uₙ = (2n² + 3n) \/ (n² - 5) lorsque n tend vers l'infini ?","option_a":"2","option_b":"3","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"En divisant numérateur et dénominateur par n², on obtient (2 + 3\/n) \/ (1 - 5\/n²) → 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76025,"question":"Soit une suite géométrique de premier terme u₀ = 5 et de raison q = -2. Quel est son terme général ?","option_a":"uₙ = 5 × (-2)ⁿ","option_b":"uₙ = 5 × 2ⁿ","option_c":"uₙ = (-2)ⁿ","option_d":"uₙ = 5 + (-2)ⁿ","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le terme général d'une suite géométrique est uₙ = u₀ × qⁿ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76026,"question":"La suite uₙ = n² - 5n + 6 est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La suite tend vers +∞ lorsque n tend vers l'infini, elle n'est donc pas bornée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76027,"question":"Quelle est la somme des 10 premiers termes d'une suite arithmétique de premier terme u₀ = 3 et de raison r = 2 ?","option_a":"120","option_b":"130","option_c":"140","option_d":"150","option_e":"","option_f":"","bonne_reponse":"B","explication":"La somme S₁₀ = (10\/2) × (2×3 + (10-1)×2) = 5 × (6 + 18) = 5 × 24 = 120.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76028,"question":"Soit la suite définie par uₙ = 1 \/ n. Quelle est sa limite lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"A","explication":"La suite tend vers 0 car 1\/n devient arbitrairement petit lorsque n devient grand.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76029,"question":"La suite uₙ = n \/ (n + 1) est-elle croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Étudions uₙ₊₁ - uₙ = (n+1)\/(n+2) - n\/(n+1) = 1\/((n+1)(n+2)) \u003E 0 pour tout n, donc la suite est croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76030,"question":"Quelle est la raison d'une suite géométrique dont le 3ème terme est 12 et le 5ème terme est 48 ?","option_a":"2","option_b":"√2","option_c":"4","option_d":"-2","option_e":"","option_f":"","bonne_reponse":"A","explication":"u₃ = u₀ × q² = 12 et u₅ = u₀ × q⁴ = 48. En divisant, on obtient q² = 4, donc q = 2 (ou -2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":76031,"question":"Soit la suite uₙ = (-1)ⁿ \/ n. Cette suite est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La suite converge vers 0 car |uₙ| = 1\/n tend vers 0 et elle alterne en signe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]