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Номер 445, страница 131 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
III. Последовательности. 18. Сумма n первых членов геометрической прогрессии - номер 445, страница 131.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108172 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-445" "field_display_title" => "445" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "18. Сумма n первых членов геометрической прогрессии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "128" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_country" => Illuminate\Database\Eloquent\Collection {#1079} "field_city" => Illuminate\Database\Eloquent\Collection {#1081} "field_series" => Illuminate\Database\Eloquent\Collection {#1083} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #original: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "18. Сумма n первых членов геометрической прогрессии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "128" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1276800 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108172" "type" => "task" ] "text" => "<p><strong>445.</strong> Дана геометрическая прогрессия ($b_n$). Заполните таблицу:</p><table><thead><tr><th></th><th>$b_1$</th><th>$q$</th><th>$n$</th><th>$b_n$</th><th>$S_n$</th></tr></thead><tbody><tr><td><strong>a)</strong></td><td>3</td><td></td><td>6</td><td>96</td><td></td></tr><tr><td><strong>б)</strong></td><td>0,2</td><td></td><td>11</td><td>204,8</td><td></td></tr><tr><td><strong>в)</strong></td><td></td><td>0,5</td><td></td><td>2</td><td>254</td></tr><tr><td><strong>г)</strong></td><td></td><td>2</td><td>8</td><td></td><td>765</td></tr><tr><td><strong>д)</strong></td><td></td><td>3</td><td></td><td>567</td><td>847</td></tr><tr><td><strong>е)</strong></td><td>2</td><td></td><td></td><td>$2^{-3}$</td><td>$3\frac{7}{8}$</td></tr></tbody></table>" "img" => array:1 [ 0 => array:5 [ "name" => "445-1.jpg" "alt" => null "width" => "1416" "height" => 875 "path" => "/media/algebra_09/soltan-u19/0-1/445-1.webp?ts=1752834463" ] ] ] #original: array:7 [ "id" => 1276800 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108172" "type" => "task" ] "text" => "<p><strong>445.</strong> Дана геометрическая прогрессия ($b_n$). Заполните таблицу:</p><table><thead><tr><th></th><th>$b_1$</th><th>$q$</th><th>$n$</th><th>$b_n$</th><th>$S_n$</th></tr></thead><tbody><tr><td><strong>a)</strong></td><td>3</td><td></td><td>6</td><td>96</td><td></td></tr><tr><td><strong>б)</strong></td><td>0,2</td><td></td><td>11</td><td>204,8</td><td></td></tr><tr><td><strong>в)</strong></td><td></td><td>0,5</td><td></td><td>2</td><td>254</td></tr><tr><td><strong>г)</strong></td><td></td><td>2</td><td>8</td><td></td><td>765</td></tr><tr><td><strong>д)</strong></td><td></td><td>3</td><td></td><td>567</td><td>847</td></tr><tr><td><strong>е)</strong></td><td>2</td><td></td><td></td><td>$2^{-3}$</td><td>$3\frac{7}{8}$</td></tr></tbody></table>" "img" => array:1 [ 0 => array:5 [ "name" => "445-1.jpg" "alt" => null "width" => "1416" "height" => 875 "path" => "/media/algebra_09/soltan-u19/0-1/445-1.webp?ts=1752834463" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1107 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278091 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108172" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "445-1.jpg" "alt" => null "width" => "1164" "height" => 1492 "path" => "/media/algebra_09/soltan-u19/1-1/445-1.webp?ts=1752831197" ] ] ] #original: array:6 [ "id" => 1278091 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108172" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "445-1.jpg" "alt" => null "width" => "1164" "height" => 1492 "path" => "/media/algebra_09/soltan-u19/1-1/445-1.webp?ts=1752831197" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1115 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554046 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108172" "type" => "task" ] "text" => "<p>Для решения задачи будем использовать следующие формулы для геометрической прогрессии:</p><p>Формула n-го члена: $b_n = b_1 \cdot q^{n-1}$</p><p>Формулы суммы первых n членов: $S_n = \frac{b_1(q^n - 1)}{q - 1}$ и $S_n = \frac{b_n q - b_1}{q - 1}$ (при $q \neq 1$)</p><br><p><strong>а)</strong></p><p>Дано: $b_1 = 3$, $n = 6$, $b_6 = 96$.<br>Найти: $q$ и $S_6$.<br>1. Найдем знаменатель прогрессии $q$ из формулы n-го члена:<br>$b_6 = b_1 \cdot q^{6-1} \Rightarrow 96 = 3 \cdot q^5$.<br>$q^5 = \frac{96}{3} = 32$, откуда $q = 2$.<br>2. Найдем сумму первых шести членов $S_6$:<br>$S_6 = \frac{b_1(q^6 - 1)}{q - 1} = \frac{3(2^6 - 1)}{2 - 1} = 3 \cdot (64 - 1) = 3 \cdot 63 = 189$.<br>Ответ: $q=2, S_6=189$.</p><p><strong>б)</strong></p><p>Дано: $b_1 = 0,2$, $n = 11$, $b_{11} = 204,8$.<br>Найти: $q$ и $S_{11}$.<br>1. Найдем знаменатель $q$:<br>$b_{11} = b_1 \cdot q^{11-1} \Rightarrow 204,8 = 0,2 \cdot q^{10}$.<br>$q^{10} = \frac{204,8}{0,2} = 1024$, откуда $q = 2$.<br>2. Найдем сумму $S_{11}$, используя формулу $S_n = \frac{b_n q - b_1}{q - 1}$:<br>$S_{11} = \frac{204,8 \cdot 2 - 0,2}{2 - 1} = 409,6 - 0,2 = 409,4$.<br>Ответ: $q=2, S_{11}=409,4$.</p><p><strong>в)</strong></p><p>Дано: $q = 0,5$, $b_n = 2$, $S_n = 254$.<br>Найти: $b_1$ и $n$.<br>1. Найдем первый член $b_1$ из формулы суммы $S_n = \frac{b_n q - b_1}{q - 1}$:<br>$254 = \frac{2 \cdot 0,5 - b_1}{0,5 - 1} \Rightarrow 254 = \frac{1 - b_1}{-0,5}$.<br>$254 \cdot (-0,5) = 1 - b_1 \Rightarrow -127 = 1 - b_1$, откуда $b_1 = 1 - (-127) = 128$.<br>2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:<br>$2 = 128 \cdot (0,5)^{n-1} \Rightarrow (0,5)^{n-1} = \frac{2}{128} = \frac{1}{64}$.<br>Так как $\frac{1}{64} = (\frac{1}{2})^6 = (0,5)^6$, то $n - 1 = 6$, откуда $n = 7$.<br>Ответ: $b_1=128, n=7$.</p><p><strong>г)</strong></p><p>Дано: $q = 2$, $n = 8$, $S_8 = 765$.<br>Найти: $b_1$ и $b_8$.<br>1. Найдем первый член $b_1$ из формулы суммы $S_n = \frac{b_1(q^n - 1)}{q - 1}$:<br>$765 = \frac{b_1(2^8 - 1)}{2 - 1} \Rightarrow 765 = b_1(256 - 1) = 255 \cdot b_1$.<br>$b_1 = \frac{765}{255} = 3$.<br>2. Найдем восьмой член $b_8$:<br>$b_8 = b_1 \cdot q^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384$.<br>Ответ: $b_1=3, b_8=384$.</p><p><strong>д)</strong></p><p>Дано: $q = 3$, $b_n = 567$, $S_n = 847$.<br>Найти: $b_1$ и $n$.<br>1. Найдем первый член $b_1$ из формулы $S_n = \frac{b_n q - b_1}{q - 1}$:<br>$847 = \frac{567 \cdot 3 - b_1}{3 - 1} \Rightarrow 847 = \frac{1701 - b_1}{2}$.<br>$847 \cdot 2 = 1701 - b_1 \Rightarrow 1694 = 1701 - b_1$, откуда $b_1 = 1701 - 1694 = 7$.<br>2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:<br>$567 = 7 \cdot 3^{n-1} \Rightarrow 3^{n-1} = \frac{567}{7} = 81$.<br>Так как $81 = 3^4$, то $n - 1 = 4$, откуда $n = 5$.<br>Ответ: $b_1=7, n=5$.</p><p><strong>е)</strong></p><p>Дано: $b_1 = 2$, $b_n = 2^{-3} = \frac{1}{8}$, $S_n = 3\frac{7}{8} = \frac{31}{8}$.<br>Найти: $q$ и $n$.<br>1. Найдем знаменатель $q$ из формулы $S_n = \frac{b_n q - b_1}{q - 1}$:<br>$\frac{31}{8} = \frac{\frac{1}{8}q - 2}{q - 1}$.<br>Умножим обе части уравнения на $8(q-1)$:<br>$31(q-1) = 8(\frac{1}{8}q - 2) \Rightarrow 31q - 31 = q - 16$.<br>$30q = 15$, откуда $q = \frac{15}{30} = \frac{1}{2}$ или $0,5$.<br>2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:<br>$\frac{1}{8} = 2 \cdot (\frac{1}{2})^{n-1} \Rightarrow \frac{1}{16} = (\frac{1}{2})^{n-1}$.<br>Так как $\frac{1}{16} = (\frac{1}{2})^4$, то $n - 1 = 4$, откуда $n = 5$.<br>Ответ: $q=\frac{1}{2}, n=5$.</p>" ] #original: array:6 [ "id" => 1554046 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108172" "type" => "task" ] "text" => "<p>Для решения задачи будем использовать следующие формулы для геометрической прогрессии:</p><p>Формула n-го члена: $b_n = b_1 \cdot q^{n-1}$</p><p>Формулы суммы первых n членов: $S_n = \frac{b_1(q^n - 1)}{q - 1}$ и $S_n = \frac{b_n q - b_1}{q - 1}$ (при $q \neq 1$)</p><br><p><strong>а)</strong></p><p>Дано: $b_1 = 3$, $n = 6$, $b_6 = 96$.<br>Найти: $q$ и $S_6$.<br>1. Найдем знаменатель прогрессии $q$ из формулы n-го члена:<br>$b_6 = b_1 \cdot q^{6-1} \Rightarrow 96 = 3 \cdot q^5$.<br>$q^5 = \frac{96}{3} = 32$, откуда $q = 2$.<br>2. Найдем сумму первых шести членов $S_6$:<br>$S_6 = \frac{b_1(q^6 - 1)}{q - 1} = \frac{3(2^6 - 1)}{2 - 1} = 3 \cdot (64 - 1) = 3 \cdot 63 = 189$.<br>Ответ: $q=2, S_6=189$.</p><p><strong>б)</strong></p><p>Дано: $b_1 = 0,2$, $n = 11$, $b_{11} = 204,8$.<br>Найти: $q$ и $S_{11}$.<br>1. Найдем знаменатель $q$:<br>$b_{11} = b_1 \cdot q^{11-1} \Rightarrow 204,8 = 0,2 \cdot q^{10}$.<br>$q^{10} = \frac{204,8}{0,2} = 1024$, откуда $q = 2$.<br>2. Найдем сумму $S_{11}$, используя формулу $S_n = \frac{b_n q - b_1}{q - 1}$:<br>$S_{11} = \frac{204,8 \cdot 2 - 0,2}{2 - 1} = 409,6 - 0,2 = 409,4$.<br>Ответ: $q=2, S_{11}=409,4$.</p><p><strong>в)</strong></p><p>Дано: $q = 0,5$, $b_n = 2$, $S_n = 254$.<br>Найти: $b_1$ и $n$.<br>1. Найдем первый член $b_1$ из формулы суммы $S_n = \frac{b_n q - b_1}{q - 1}$:<br>$254 = \frac{2 \cdot 0,5 - b_1}{0,5 - 1} \Rightarrow 254 = \frac{1 - b_1}{-0,5}$.<br>$254 \cdot (-0,5) = 1 - b_1 \Rightarrow -127 = 1 - b_1$, откуда $b_1 = 1 - (-127) = 128$.<br>2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:<br>$2 = 128 \cdot (0,5)^{n-1} \Rightarrow (0,5)^{n-1} = \frac{2}{128} = \frac{1}{64}$.<br>Так как $\frac{1}{64} = (\frac{1}{2})^6 = (0,5)^6$, то $n - 1 = 6$, откуда $n = 7$.<br>Ответ: $b_1=128, n=7$.</p><p><strong>г)</strong></p><p>Дано: $q = 2$, $n = 8$, $S_8 = 765$.<br>Найти: $b_1$ и $b_8$.<br>1. Найдем первый член $b_1$ из формулы суммы $S_n = \frac{b_1(q^n - 1)}{q - 1}$:<br>$765 = \frac{b_1(2^8 - 1)}{2 - 1} \Rightarrow 765 = b_1(256 - 1) = 255 \cdot b_1$.<br>$b_1 = \frac{765}{255} = 3$.<br>2. Найдем восьмой член $b_8$:<br>$b_8 = b_1 \cdot q^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384$.<br>Ответ: $b_1=3, b_8=384$.</p><p><strong>д)</strong></p><p>Дано: $q = 3$, $b_n = 567$, $S_n = 847$.<br>Найти: $b_1$ и $n$.<br>1. Найдем первый член $b_1$ из формулы $S_n = \frac{b_n q - b_1}{q - 1}$:<br>$847 = \frac{567 \cdot 3 - b_1}{3 - 1} \Rightarrow 847 = \frac{1701 - b_1}{2}$.<br>$847 \cdot 2 = 1701 - b_1 \Rightarrow 1694 = 1701 - b_1$, откуда $b_1 = 1701 - 1694 = 7$.<br>2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:<br>$567 = 7 \cdot 3^{n-1} \Rightarrow 3^{n-1} = \frac{567}{7} = 81$.<br>Так как $81 = 3^4$, то $n - 1 = 4$, откуда $n = 5$.<br>Ответ: $b_1=7, n=5$.</p><p><strong>е)</strong></p><p>Дано: $b_1 = 2$, $b_n = 2^{-3} = \frac{1}{8}$, $S_n = 3\frac{7}{8} = \frac{31}{8}$.<br>Найти: $q$ и $n$.<br>1. Найдем знаменатель $q$ из формулы $S_n = \frac{b_n q - b_1}{q - 1}$:<br>$\frac{31}{8} = \frac{\frac{1}{8}q - 2}{q - 1}$.<br>Умножим обе части уравнения на $8(q-1)$:<br>$31(q-1) = 8(\frac{1}{8}q - 2) \Rightarrow 31q - 31 = q - 16$.<br>$30q = 15$, откуда $q = \frac{15}{30} = \frac{1}{2}$ или $0,5$.<br>2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:<br>$\frac{1}{8} = 2 \cdot (\frac{1}{2})^{n-1} \Rightarrow \frac{1}{16} = (\frac{1}{2})^{n-1}$.<br>Так как $\frac{1}{16} = (\frac{1}{2})^4$, то $n - 1 = 4$, откуда $n = 5$.<br>Ответ: $q=\frac{1}{2}, n=5$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108173" "type" => "task" ] "previous" => array:2 [ "refs" => "1108171" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true 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"field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1969 #items: array:1 [ 0 => App\Models\BookPage {#1155 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097873 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-131" "field_display_title" => "131" "field_folder" => "folder1" "field_image_name" => "131" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Book {#1160 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Element {#1190 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1261024 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261024 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097874" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097872" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:6 [ 0 => App\Models\Task {#1199 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108172 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-445" "field_display_title" => "445" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1298 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108172 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-445" "field_display_title" => "445" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1298 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1327 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108173 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-446" "field_display_title" => "446" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1330 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1331 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1364 …2} "content" => Illuminate\Database\Eloquent\Collection {#1401 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1426 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108173 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-446" "field_display_title" => "446" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1330 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1331 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1364 …2} "content" => Illuminate\Database\Eloquent\Collection {#1401 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1426 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1455 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108174 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-447" "field_display_title" => "447" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1456 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1458 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1459 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1492 …2} "content" => Illuminate\Database\Eloquent\Collection {#1529 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1554 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108174 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-447" "field_display_title" => "447" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1456 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1458 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1459 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1492 …2} "content" => Illuminate\Database\Eloquent\Collection {#1529 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1554 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1583 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108175 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-448" "field_display_title" => "448" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1584 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1586 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1587 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1620 …2} "content" => Illuminate\Database\Eloquent\Collection {#1657 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1682 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108175 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-448" "field_display_title" => "448" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1584 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1586 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1587 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1620 …2} "content" => Illuminate\Database\Eloquent\Collection {#1657 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1682 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1711 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108176 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-449" "field_display_title" => "449" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1712 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1714 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1715 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1748 …2} "content" => Illuminate\Database\Eloquent\Collection {#1785 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1810 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108176 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "131" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-449" "field_display_title" => "449" "field_outside_task" => null "field_task_type" => 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+usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№445 (с. 131)
Условие. №445 (с. 131)
скриншот условия
445. Дана геометрическая прогрессия ($b_n$). Заполните таблицу:
| $b_1$ | $q$ | $n$ | $b_n$ | $S_n$ | |
|---|---|---|---|---|---|
| a) | 3 | 6 | 96 | ||
| б) | 0,2 | 11 | 204,8 | ||
| в) | 0,5 | 2 | 254 | ||
| г) | 2 | 8 | 765 | ||
| д) | 3 | 567 | 847 | ||
| е) | 2 | $2^{-3}$ | $3\frac{7}{8}$ |
Решение. №445 (с. 131)
Решение 2 (rus). №445 (с. 131)
Для решения задачи будем использовать следующие формулы для геометрической прогрессии:
Формула n-го члена: $b_n = b_1 \cdot q^{n-1}$
Формулы суммы первых n членов: $S_n = \frac{b_1(q^n - 1)}{q - 1}$ и $S_n = \frac{b_n q - b_1}{q - 1}$ (при $q \neq 1$)
а)
Дано: $b_1 = 3$, $n = 6$, $b_6 = 96$.
Найти: $q$ и $S_6$.
1. Найдем знаменатель прогрессии $q$ из формулы n-го члена:
$b_6 = b_1 \cdot q^{6-1} \Rightarrow 96 = 3 \cdot q^5$.
$q^5 = \frac{96}{3} = 32$, откуда $q = 2$.
2. Найдем сумму первых шести членов $S_6$:
$S_6 = \frac{b_1(q^6 - 1)}{q - 1} = \frac{3(2^6 - 1)}{2 - 1} = 3 \cdot (64 - 1) = 3 \cdot 63 = 189$.
Ответ: $q=2, S_6=189$.
б)
Дано: $b_1 = 0,2$, $n = 11$, $b_{11} = 204,8$.
Найти: $q$ и $S_{11}$.
1. Найдем знаменатель $q$:
$b_{11} = b_1 \cdot q^{11-1} \Rightarrow 204,8 = 0,2 \cdot q^{10}$.
$q^{10} = \frac{204,8}{0,2} = 1024$, откуда $q = 2$.
2. Найдем сумму $S_{11}$, используя формулу $S_n = \frac{b_n q - b_1}{q - 1}$:
$S_{11} = \frac{204,8 \cdot 2 - 0,2}{2 - 1} = 409,6 - 0,2 = 409,4$.
Ответ: $q=2, S_{11}=409,4$.
в)
Дано: $q = 0,5$, $b_n = 2$, $S_n = 254$.
Найти: $b_1$ и $n$.
1. Найдем первый член $b_1$ из формулы суммы $S_n = \frac{b_n q - b_1}{q - 1}$:
$254 = \frac{2 \cdot 0,5 - b_1}{0,5 - 1} \Rightarrow 254 = \frac{1 - b_1}{-0,5}$.
$254 \cdot (-0,5) = 1 - b_1 \Rightarrow -127 = 1 - b_1$, откуда $b_1 = 1 - (-127) = 128$.
2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:
$2 = 128 \cdot (0,5)^{n-1} \Rightarrow (0,5)^{n-1} = \frac{2}{128} = \frac{1}{64}$.
Так как $\frac{1}{64} = (\frac{1}{2})^6 = (0,5)^6$, то $n - 1 = 6$, откуда $n = 7$.
Ответ: $b_1=128, n=7$.
г)
Дано: $q = 2$, $n = 8$, $S_8 = 765$.
Найти: $b_1$ и $b_8$.
1. Найдем первый член $b_1$ из формулы суммы $S_n = \frac{b_1(q^n - 1)}{q - 1}$:
$765 = \frac{b_1(2^8 - 1)}{2 - 1} \Rightarrow 765 = b_1(256 - 1) = 255 \cdot b_1$.
$b_1 = \frac{765}{255} = 3$.
2. Найдем восьмой член $b_8$:
$b_8 = b_1 \cdot q^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384$.
Ответ: $b_1=3, b_8=384$.
д)
Дано: $q = 3$, $b_n = 567$, $S_n = 847$.
Найти: $b_1$ и $n$.
1. Найдем первый член $b_1$ из формулы $S_n = \frac{b_n q - b_1}{q - 1}$:
$847 = \frac{567 \cdot 3 - b_1}{3 - 1} \Rightarrow 847 = \frac{1701 - b_1}{2}$.
$847 \cdot 2 = 1701 - b_1 \Rightarrow 1694 = 1701 - b_1$, откуда $b_1 = 1701 - 1694 = 7$.
2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:
$567 = 7 \cdot 3^{n-1} \Rightarrow 3^{n-1} = \frac{567}{7} = 81$.
Так как $81 = 3^4$, то $n - 1 = 4$, откуда $n = 5$.
Ответ: $b_1=7, n=5$.
е)
Дано: $b_1 = 2$, $b_n = 2^{-3} = \frac{1}{8}$, $S_n = 3\frac{7}{8} = \frac{31}{8}$.
Найти: $q$ и $n$.
1. Найдем знаменатель $q$ из формулы $S_n = \frac{b_n q - b_1}{q - 1}$:
$\frac{31}{8} = \frac{\frac{1}{8}q - 2}{q - 1}$.
Умножим обе части уравнения на $8(q-1)$:
$31(q-1) = 8(\frac{1}{8}q - 2) \Rightarrow 31q - 31 = q - 16$.
$30q = 15$, откуда $q = \frac{15}{30} = \frac{1}{2}$ или $0,5$.
2. Найдем номер члена $n$ из формулы $b_n = b_1 \cdot q^{n-1}$:
$\frac{1}{8} = 2 \cdot (\frac{1}{2})^{n-1} \Rightarrow \frac{1}{16} = (\frac{1}{2})^{n-1}$.
Так как $\frac{1}{16} = (\frac{1}{2})^4$, то $n - 1 = 4$, откуда $n = 5$.
Ответ: $q=\frac{1}{2}, n=5$.
Другие задания:
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