Maîtrisez les suites réelles : QCM interactif pour la 3ème année secondaire

Question 1 sur 10 10:00

[{"id":34283,"question":"Quelle est la limite de la suite définie par uₙ = (2n + 1)\/(3n - 5) lorsque n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 2\/3","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"On divise numérateur et dénominateur par n : uₙ = (2 + 1\/n)\/(3 - 5\/n). Lorsque n → +∞, 1\/n → 0, donc uₙ → 2\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34284,"question":"Une suite croissante et non majorée est toujours divergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une suite croissante et non majorée diverge vers +∞, donc elle est divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34285,"question":"Soit la suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 3. Quelle est l'expression de uₙ en fonction de n ?","option_a":"A. uₙ = 2ⁿ + 3","option_b":"B. uₙ = 2ⁿ⁺¹ - 3","option_c":"C. uₙ = 3·2ⁿ - 2","option_d":"D. uₙ = 2·3ⁿ - 1","option_e":"","option_f":"","bonne_reponse":"C","explication":"En résolvant la récurrence linéaire, on trouve uₙ = 3·2ⁿ - 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34286,"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, donc elle n'a pas de limite finie et est divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34287,"question":"Quelle est la limite de la suite uₙ = n² - 5n + 1 lorsque n tend vers +∞ ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. -∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le terme dominant est n², donc uₙ → +∞ lorsque n → +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34288,"question":"Soit une suite géométrique de premier terme u₀ = 5 et de raison q = 1\/2. Quelle est la limite de cette suite ?","option_a":"A. 0","option_b":"B. 5","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Puisque |q| = 1\/2 \u003C 1, la suite converge vers 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34289,"question":"La suite définie par uₙ = sin(n) est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction sinus est bornée entre -1 et 1, donc la suite uₙ est bornée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34290,"question":"Quelle est la nature de la suite définie par uₙ = (n + 1)\/(n + 2) ?","option_a":"A. Croissante","option_b":"B. Décroissante","option_c":"C. Constante","option_d":"D. Non monotone","option_e":"","option_f":"","bonne_reponse":"B","explication":"En comparant uₙ et uₙ₊₁, on montre que uₙ₊₁ \u003C uₙ, donc la suite est décroissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34291,"question":"Soit une suite arithmétique de premier terme u₀ = 3 et de raison r = -2. Quel est le terme u₅ ?","option_a":"A. -7","option_b":"B. 7","option_c":"C. -5","option_d":"D. 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"u₅ = u₀ + 5r = 3 + 5×(-2) = 3 - 10 = -7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":34292,"question":"Toute suite convergente est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Si une suite converge vers une limite finie, elle est nécessairement bornée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]