Quiz — Inégalités et récurrences.pdf

Question 1 sur 10 20:00

[{"id":83447,"question":"Quelle est la première étape d’une démonstration par récurrence ?","option_a":"Vérifier l’hypothèse de récurrence","option_b":"Initialiser la propriété pour n=0","option_c":"Conclure directement","option_d":"Montrer que la propriété est vraie pour tout n","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’initialisation est cruciale : elle consiste à vérifier que la propriété est vraie pour le premier terme (souvent n=0 ou n=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\": \"Vérifier l’hypothèse de récurrence\", \"b\": \"Initialiser la pr","_debug_options_count":4},{"id":83448,"question":"L’inégalité de Bernoulli s’écrit : (1 + x)^n ≥ 1 + nx pour x ≥ -1 et n ∈ ℕ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette inégalité est vraie pour x ≥ -1 et n ∈ ℕ, mais elle est souvent utilisée pour x \u003E -1 dans les exercices.","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":83449,"question":"Soit la suite définie par u_{n+1} = 2u_n + 1 avec u_0 = 0. Quelle est l’expression de u_n ?","option_a":"u_n = 2^n - 1","option_b":"u_n = 2^n + 1","option_c":"u_n = n^2","option_d":"u_n = 2n + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution est u_n = 2^n - 1, obtenue en résolvant l’équation de récurrence linéaire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"u_n = 2^n - 1\", \"b\": \"u_n = 2^n + 1\", \"c\": \"u_n = n^2\", \"d\": \"u_n","_debug_options_count":4},{"id":83450,"question":"L’inégalité triangulaire stipule que pour tous réels a et b, |a + b| ≤ |a| + |b|.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette inégalité est fondamentale en analyse et s’applique à tous les réels a et b.","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":83451,"question":"Soit une suite (u_n) croissante et majorée. Que peut-on conclure ?","option_a":"Elle converge","option_b":"Elle diverge","option_c":"Elle est constante","option_d":"Elle est périodique","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute suite croissante et majorée converge vers sa borne supérieure (théorème de convergence monotone).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Elle converge\", \"b\": \"Elle diverge\", \"c\": \"Elle est constante\", \"","_debug_options_count":4},{"id":83452,"question":"Quelle est la limite de la suite u_n = (1 + 1\/n)^n quand n tend vers l’infini ?","option_a":"0","option_b":"1","option_c":"e","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Cette suite converge vers le nombre d’Euler e ≈ 2,71828, base du logarithme naturel.","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\": \"e\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":83453,"question":"L’inégalité de Cauchy-Schwarz s’applique aux vecteurs de ℝ^n.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Elle stipule que |⟨u, v⟩| ≤ ||u|| ||v|| pour tous vecteurs u et v de ℝ^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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":83454,"question":"Soit la suite définie par u_{n+1} = u_n^2 - 2 avec u_0 = 2. Que vaut u_2 ?","option_a":"2","option_b":"0","option_c":"4","option_d":"-2","option_e":"","option_f":"","bonne_reponse":"b","explication":"u_1 = 2^2 - 2 = 2, u_2 = 2^2 - 2 = 2. La suite est constante à partir de u_0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"0\", \"c\": \"4\", \"d\": \"-2\"}}","_debug_options_count":4},{"id":83455,"question":"Quelle méthode permet de résoudre une inégalité du type f(x) ≥ g(x) ?","option_a":"Étudier le signe de f(x) - g(x)","option_b":"Calculer f(x) + g(x)","option_c":"Résoudre f(x) = g(x) puis tester des valeurs","option_d":"Tracer les courbes de f et g","option_e":"","option_f":"","bonne_reponse":"a","explication":"Étudier le signe de f(x) - g(x) permet de déterminer où f(x) ≥ g(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Étudier le signe de f(x) - g(x)\", \"b\": \"Calculer f(x) + g(x)\", \"","_debug_options_count":4},{"id":83456,"question":"La suite u_n = (-1)^n est-elle 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 et donc ne converge pas.","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}]