Quiz — Quelques calculs de sommes.pdf

Question 1 sur 10 20:00

[{"id":90501,"question":"Quelle est la valeur de la somme S = 1 + 2 + 3 + ... + n ?","option_a":"n(n+1)\/2","option_b":"n(n-1)\/2","option_c":"n²","option_d":"2n","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de la somme des n premiers entiers naturels est S = n(n+1)\/2.","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(n+1)\/2\", \"b\": \"n(n-1)\/2\", \"c\": \"n²\", \"d\": \"2n\"}}","_debug_options_count":4},{"id":90502,"question":"Une suite géométrique de premier terme u₀ = 3 et de raison r = 2 a pour somme partielle S₅ = ?","option_a":"93","option_b":"96","option_c":"90","option_d":"102","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme partielle d'une suite géométrique est Sₙ = u₀(1 - r^{n+1})\/(1 - r). Ici, S₅ = 3(1 - 2⁶)\/(1 - 2) = 3(1 - 64)\/(-1) = 189.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"93\", \"b\": \"96\", \"c\": \"90\", \"d\": \"102\"}}","_debug_options_count":4},{"id":90503,"question":"La série Σ (de n=1 à ∞) 1\/n² est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série Σ 1\/n² est une série de Riemann convergente (p = 2 \u003E 1).","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":90504,"question":"Quelle est la somme des 4 premiers termes d'une suite arithmétique de premier terme u₁ = 5 et de raison r = 3 ?","option_a":"26","option_b":"30","option_c":"32","option_d":"28","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les termes sont 5, 8, 11, 14. Leur somme est 5 + 8 + 11 + 14 = 38. La formule Sₙ = n(u₁ + uₙ)\/2 donne S₄ = 4(5 + 14)\/2 = 38.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"26\", \"b\": \"30\", \"c\": \"32\", \"d\": \"28\"}}","_debug_options_count":4},{"id":90505,"question":"La somme Σ (de k=1 à n) k(k+1) peut s'écrire sous la forme :","option_a":"n(n+1)(n+2)\/3","option_b":"n(n+1)(2n+1)\/6","option_c":"n²(n+1)²\/4","option_d":"n(n+2)\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En développant k(k+1) = k² + k, on utilise les formules Σk = n(n+1)\/2 et Σk² = n(n+1)(2n+1)\/6. La somme totale est n(n+1)(n+2)\/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(n+1)(n+2)\/3\", \"b\": \"n(n+1)(2n+1)\/6\", \"c\": \"n²(n+1)²\/4\", \"d\": ","_debug_options_count":4},{"id":90506,"question":"Une série télescopique est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une série télescopique est convergente uniquement si la limite de son terme général est nulle.","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":90507,"question":"Quelle est la somme S = Σ (de k=0 à 4) 2^k ?","option_a":"15","option_b":"31","option_c":"16","option_d":"30","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une somme géométrique de raison 2 et de 5 termes. S = 2⁵ - 1 = 31.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"15\", \"b\": \"31\", \"c\": \"16\", \"d\": \"30\"}}","_debug_options_count":4},{"id":90508,"question":"La somme Σ (de n=1 à ∞) 1\/n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série harmonique Σ 1\/n est divergente (série de Riemann avec p = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":90509,"question":"Quelle formule permet de calculer la somme d'une suite arithmétique de n termes ?","option_a":"Sₙ = n(u₁ + uₙ)\/2","option_b":"Sₙ = u₁(1 - rⁿ)\/(1 - r)","option_c":"Sₙ = n²","option_d":"Sₙ = u₁ + uₙ","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de la somme d'une suite arithmétique est Sₙ = n(u₁ + uₙ)\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Sₙ = n(u₁ + uₙ)\/2\", \"b\": \"Sₙ = u₁(1 - rⁿ)\/(1 - r)\", \"","_debug_options_count":4},{"id":90510,"question":"Si une suite est définie par uₙ = 3n + 2, quelle est la somme de ses 10 premiers termes ?","option_a":"175","option_b":"185","option_c":"165","option_d":"195","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une suite arithmétique de raison 3 et de premier terme u₁ = 5. S₁₀ = 10(5 + 32)\/2 = 185.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"175\", \"b\": \"185\", \"c\": \"165\", \"d\": \"195\"}}","_debug_options_count":4}]