Quiz — سلسلة في المتتاليات الهندسية.pdf

Question 1 sur 10 20:00

[{"id":17939,"question":"Quelle est la formule du terme général d'une suite géométrique de raison q et de premier terme u_1 ?","option_a":"u_n = u_1 + (n-1)q","option_b":"u_n = u_1 * q^(n-1)","option_c":"u_n = u_1 * n * q","option_d":"u_n = (u_1 + q)^n","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule correcte est u_n = u_1 * q^(n-1), où q est la raison et u_1 le premier terme.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"u_n = u_1 + (n-1)q\", \"b\": \"u_n = u_1 * q^(n-1)\", \"c\": \"u_n = u_1 ","_debug_options_count":4},{"id":17940,"question":"Si une suite géométrique a pour raison q = 2, alors elle est toujours croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. La suite est croissante si u_1 \u003E 0 et q \u003E 1, mais décroissante si u_1 \u003C 0 et q \u003E 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":17941,"question":"Quelle est la somme des 5 premiers termes d'une suite géométrique de raison q = 1\/2 et de premier terme u_1 = 32 ?","option_a":"62","option_b":"63","option_c":"64","option_d":"65","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme S_5 = 32 * (1 - (1\/2)^5) \/ (1 - 1\/2) = 32 * (31\/32) \/ (1\/2) = 62.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"62\", \"b\": \"63\", \"c\": \"64\", \"d\": \"65\"}}","_debug_options_count":4},{"id":17942,"question":"Une suite géométrique de raison q = -0.5 converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La suite converge car |q| = 0.5 \u003C 1, même si q est négatif.","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":17943,"question":"Quel est le premier terme d'une suite géométrique de raison q = 3 et dont le 4ème terme est u_4 = 54 ?","option_a":"2","option_b":"3","option_c":"6","option_d":"9","option_e":"","option_f":"","bonne_reponse":"b","explication":"On a u_4 = u_1 * 3^(4-1) = u_1 * 27 = 54, donc u_1 = 54 \/ 27 = 2.","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\": \"3\", \"c\": \"6\", \"d\": \"9\"}}","_debug_options_count":4},{"id":17944,"question":"La suite géométrique définie par u_n = 5 * (-2)^n est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Cette suite n'est pas bornée car ses termes tendent vers l'infini en valeur absolue.","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":17945,"question":"Quelle est la raison d'une suite géométrique dont les termes sont : 8, -4, 2, -1, 0.5 ?","option_a":"-0.5","option_b":"0.5","option_c":"-2","option_d":"2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La raison q est obtenue en divisant un terme par le précédent : -4 \/ 8 = -0.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-0.5\", \"b\": \"0.5\", \"c\": \"-2\", \"d\": \"2\"}}","_debug_options_count":4},{"id":17946,"question":"Si une suite géométrique a pour premier terme u_1 = 10 et pour raison q = 0.1, que vaut la limite de u_n quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"10","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite est 0 car |q| = 0.1 \u003C 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\": \"0\", \"b\": \"1\", \"c\": \"10\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":17947,"question":"La somme infinie d'une suite géométrique de raison q = 0.25 existe-t-elle ? Si oui, quelle est sa valeur pour u_1 = 4 ?","option_a":"Non, elle n'existe pas","option_b":"Oui, S = 5.33","option_c":"Oui, S = 16","option_d":"Oui, S = 4.5","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme infinie existe car |q| \u003C 1. S = u_1 \/ (1 - q) = 4 \/ 0.75 ≈ 5.33.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Non, elle n'existe pas\", \"b\": \"Oui, S = 5.33\", \"c\": \"Oui, S = 16\"","_debug_options_count":4},{"id":17948,"question":"Une suite géométrique de raison q = 1 est-elle constante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Si q = 1, alors u_n = u_1 pour tout n, donc la suite est constante.","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}]