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Номер 384, страница 114 - 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. 15. Арифметическая профессия и ее свойства - номер 384, страница 114.
App\Models\Task {#1030 // 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" => 1108108 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-384" "field_display_title" => "384" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #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 {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #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 {#1039 #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 {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #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 {#1041 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 #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 {#1064 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 #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 {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045} "field_author" => Illuminate\Database\Eloquent\Collection {#1047} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052} "field_country" => Illuminate\Database\Eloquent\Collection {#1054} "field_city" => Illuminate\Database\Eloquent\Collection {#1056} "field_series" => Illuminate\Database\Eloquent\Collection {#1058} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063} "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 {#1064} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067} "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 {#1068 #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 {#1039} "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 {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1104 #items: array:2 [ 0 => App\Models\Branch {#1112 #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 {#1113 #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 {#1144 #items: array:1 [ 0 => App\Models\Book {#1114 #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 {#1116 #items: array:1 [ 0 => App\Models\Term {#1115 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" …3 ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #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_class" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Term {#1118 #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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #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_publisher" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 #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" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #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_author" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:3 [ 0 => App\Models\Term {#1122 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #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\Term {#1123 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #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\Term {#1124 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => 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 } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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_country" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 #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" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #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_city" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1131 #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" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #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_series" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137 #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 [ "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 {#1138 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141 #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 {#1116} "field_class" => Illuminate\Database\Eloquent\Collection {#1117} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1119} "field_author" => Illuminate\Database\Eloquent\Collection {#1121} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1125} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_country" => Illuminate\Database\Eloquent\Collection {#1128} "field_city" => Illuminate\Database\Eloquent\Collection {#1130} "field_series" => Illuminate\Database\Eloquent\Collection {#1132} "field_umk" => Illuminate\Database\Eloquent\Collection {#1133} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137} "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 [ "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 {#1138} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141} "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 {#1142 #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 {#1113} "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 {#1144} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #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\Branch {#1143 #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" => 1107656 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "15. Арифметическая профессия и ее свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "108" "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 {#1146 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107656 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "15. Арифметическая профессия и ее свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145} "field_page_start" => "108" "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 {#1146} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147} ] #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 {#1101 #items: array:3 [ 0 => App\Models\Element {#1094 #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" => 1276739 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Edition {#1086 #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 {#1075 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1076 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "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 {#1075} "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 {#1076} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080} "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" => "1108108" "type" => "task" ] "text" => "<p><strong>384.</strong> Найдите первый член и разность арифметической прогрессии $a_n$, если:</p><p><strong>а)</strong> $a_2 + a_4 + a_6 = 18$ и $a_2 \cdot a_4 \cdot a_6 = 120;$</p><p><strong>б)</strong> $a_1 + a_2 + a_3 = -12$ и $a_1^2 + a_2^2 + a_3^2 = 98.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "384-1.jpg" "alt" => null "width" => "1553" "height" => 345 "path" => "/media/algebra_09/soltan-u19/0-1/384-1.webp?ts=1752834418" ] ] ] #original: array:7 [ "id" => 1276739 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1084} "task" => array:2 [ "refs" => "1108108" "type" => "task" ] "text" => "<p><strong>384.</strong> Найдите первый член и разность арифметической прогрессии $a_n$, если:</p><p><strong>а)</strong> $a_2 + a_4 + a_6 = 18$ и $a_2 \cdot a_4 \cdot a_6 = 120;$</p><p><strong>б)</strong> $a_1 + a_2 + a_3 = -12$ и $a_1^2 + a_2^2 + a_3^2 = 98.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "384-1.jpg" "alt" => null "width" => "1553" "height" => 345 "path" => "/media/algebra_09/soltan-u19/0-1/384-1.webp?ts=1752834418" ] ] ] #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 {#1092 #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" => 1278030 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ 0 => App\Models\Edition {#1091 #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 {#1087 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1089 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "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 {#1087} "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 {#1089} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090} "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" => "1108108" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "384-1.jpg" "alt" => null "width" => "1255" "height" => 1722 "path" => "/media/algebra_09/soltan-u19/1-1/384-1.webp?ts=1752831142" ] 1 => array:5 [ "name" => "384-2.jpg" "alt" => null "width" => "1323" "height" => 1450 "path" => "/media/algebra_09/soltan-u19/1-1/384-2.webp?ts=1752831142" ] ] ] #original: array:6 [ "id" => 1278030 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1078} "task" => array:2 [ "refs" => "1108108" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "384-1.jpg" "alt" => null "width" => "1255" "height" => 1722 "path" => "/media/algebra_09/soltan-u19/1-1/384-1.webp?ts=1752831142" ] 1 => array:5 [ "name" => "384-2.jpg" "alt" => null "width" => "1323" "height" => 1450 "path" => "/media/algebra_09/soltan-u19/1-1/384-2.webp?ts=1752831142" ] ] ] #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 {#1083 #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" => 1553985 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Edition {#1079 #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 {#1072 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1103 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1111 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } "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 {#1072} "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 {#1103} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1111} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1107} "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" => "1108108" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Даны условия для арифметической прогрессии $(a_n)$: $a_2 + a_4 + a_6 = 18$ и $a_2 \cdot a_4 \cdot a_6 = 120$.</p><p>Воспользуемся формулой n-го члена арифметической прогрессии $a_n = a_1 + (n-1)d$, где $a_1$ — первый член, а $d$ — разность прогрессии. Выразим члены $a_2, a_4, a_6$ через $a_1$ и $d$:</p><p>$a_2 = a_1 + d$</p><p>$a_4 = a_1 + 3d$</p><p>$a_6 = a_1 + 5d$</p><p>Подставим эти выражения в первое уравнение:</p><p>$(a_1 + d) + (a_1 + 3d) + (a_1 + 5d) = 18$</p><p>$3a_1 + 9d = 18$</p><p>Разделив обе части на 3, получим: $a_1 + 3d = 6$.</p><p>Левая часть этого уравнения, $a_1 + 3d$, является формулой для $a_4$, следовательно, $a_4 = 6$.</p><p>Теперь подставим $a_4 = 6$ во второе уравнение:</p><p>$a_2 \cdot 6 \cdot a_6 = 120$, откуда $a_2 \cdot a_6 = 20$.</p><p>Выразим $a_2$ и $a_6$ через $a_4$ и $d$:</p><p>$a_2 = a_4 - 2d = 6 - 2d$</p><p>$a_6 = a_4 + 2d = 6 + 2d$</p><p>Подставим эти выражения в уравнение $a_2 \cdot a_6 = 20$:</p><p>$(6 - 2d)(6 + 2d) = 20$</p><p>$36 - 4d^2 = 20$</p><p>$4d^2 = 16$</p><p>$d^2 = 4$</p><p>Отсюда получаем два возможных значения для разности: $d = 2$ и $d = -2$.</p><p>Теперь найдем $a_1$ для каждого значения $d$, используя уравнение $a_1 = 6 - 3d$.</p><p><strong>1.</strong> Если $d = 2$, то $a_1 = 6 - 3(2) = 0$.</p><p><strong>2.</strong> Если $d = -2$, то $a_1 = 6 - 3(-2) = 12$.</p><p>Таким образом, существуют две прогрессии, удовлетворяющие заданным условиям.</p><p><strong>Ответ:</strong> $a_1 = 0, d = 2$ или $a_1 = 12, d = -2$.</p><p><strong>б)</strong></p><p>Даны условия: $a_1 + a_2 + a_3 = -12$ и $a_1^2 + a_2^2 + a_3^2 = 98$.</p><p>Выразим члены прогрессии $a_2$ и $a_3$ через $a_1$ и $d$:</p><p>$a_2 = a_1 + d$</p><p>$a_3 = a_1 + 2d$</p><p>Подставим в первое уравнение:</p><p>$a_1 + (a_1 + d) + (a_1 + 2d) = -12$</p><p>$3a_1 + 3d = -12$</p><p>Разделив обе части на 3, получим: $a_1 + d = -4$.</p><p>Левая часть этого уравнения, $a_1 + d$, является формулой для $a_2$, следовательно, $a_2 = -4$.</p><p>Теперь выразим $a_1$ и $a_3$ через $a_2$ и $d$:</p><p>$a_1 = a_2 - d = -4 - d$</p><p>$a_3 = a_2 + d = -4 + d$</p><p>Подставим эти выражения и значение $a_2 = -4$ во второе уравнение:</p><p>$(-4 - d)^2 + (-4)^2 + (-4 + d)^2 = 98$</p><p>Раскроем скобки и упростим:</p><p>$(16 + 8d + d^2) + 16 + (16 - 8d + d^2) = 98$</p><p>$48 + 2d^2 = 98$</p><p>$2d^2 = 50$</p><p>$d^2 = 25$</p><p>Отсюда получаем два возможных значения для разности: $d = 5$ и $d = -5$.</p><p>Найдем $a_1$ для каждого случая, используя $a_1 = -4 - d$.</p><p><strong>1.</strong> Если $d = 5$, то $a_1 = -4 - 5 = -9$.</p><p><strong>2.</strong> Если $d = -5$, то $a_1 = -4 - (-5) = 1$.</p><p>Таким образом, существуют две прогрессии, удовлетворяющие заданным условиям.</p><p><strong>Ответ:</strong> $a_1 = -9, d = 5$ или $a_1 = 1, d = -5$.</p>" ] #original: array:6 [ "id" => 1553985 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1106} "task" => array:2 [ "refs" => "1108108" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Даны условия для арифметической прогрессии $(a_n)$: $a_2 + a_4 + a_6 = 18$ и $a_2 \cdot a_4 \cdot a_6 = 120$.</p><p>Воспользуемся формулой n-го члена арифметической прогрессии $a_n = a_1 + (n-1)d$, где $a_1$ — первый член, а $d$ — разность прогрессии. Выразим члены $a_2, a_4, a_6$ через $a_1$ и $d$:</p><p>$a_2 = a_1 + d$</p><p>$a_4 = a_1 + 3d$</p><p>$a_6 = a_1 + 5d$</p><p>Подставим эти выражения в первое уравнение:</p><p>$(a_1 + d) + (a_1 + 3d) + (a_1 + 5d) = 18$</p><p>$3a_1 + 9d = 18$</p><p>Разделив обе части на 3, получим: $a_1 + 3d = 6$.</p><p>Левая часть этого уравнения, $a_1 + 3d$, является формулой для $a_4$, следовательно, $a_4 = 6$.</p><p>Теперь подставим $a_4 = 6$ во второе уравнение:</p><p>$a_2 \cdot 6 \cdot a_6 = 120$, откуда $a_2 \cdot a_6 = 20$.</p><p>Выразим $a_2$ и $a_6$ через $a_4$ и $d$:</p><p>$a_2 = a_4 - 2d = 6 - 2d$</p><p>$a_6 = a_4 + 2d = 6 + 2d$</p><p>Подставим эти выражения в уравнение $a_2 \cdot a_6 = 20$:</p><p>$(6 - 2d)(6 + 2d) = 20$</p><p>$36 - 4d^2 = 20$</p><p>$4d^2 = 16$</p><p>$d^2 = 4$</p><p>Отсюда получаем два возможных значения для разности: $d = 2$ и $d = -2$.</p><p>Теперь найдем $a_1$ для каждого значения $d$, используя уравнение $a_1 = 6 - 3d$.</p><p><strong>1.</strong> Если $d = 2$, то $a_1 = 6 - 3(2) = 0$.</p><p><strong>2.</strong> Если $d = -2$, то $a_1 = 6 - 3(-2) = 12$.</p><p>Таким образом, существуют две прогрессии, удовлетворяющие заданным условиям.</p><p><strong>Ответ:</strong> $a_1 = 0, d = 2$ или $a_1 = 12, d = -2$.</p><p><strong>б)</strong></p><p>Даны условия: $a_1 + a_2 + a_3 = -12$ и $a_1^2 + a_2^2 + a_3^2 = 98$.</p><p>Выразим члены прогрессии $a_2$ и $a_3$ через $a_1$ и $d$:</p><p>$a_2 = a_1 + d$</p><p>$a_3 = a_1 + 2d$</p><p>Подставим в первое уравнение:</p><p>$a_1 + (a_1 + d) + (a_1 + 2d) = -12$</p><p>$3a_1 + 3d = -12$</p><p>Разделив обе части на 3, получим: $a_1 + d = -4$.</p><p>Левая часть этого уравнения, $a_1 + d$, является формулой для $a_2$, следовательно, $a_2 = -4$.</p><p>Теперь выразим $a_1$ и $a_3$ через $a_2$ и $d$:</p><p>$a_1 = a_2 - d = -4 - d$</p><p>$a_3 = a_2 + d = -4 + d$</p><p>Подставим эти выражения и значение $a_2 = -4$ во второе уравнение:</p><p>$(-4 - d)^2 + (-4)^2 + (-4 + d)^2 = 98$</p><p>Раскроем скобки и упростим:</p><p>$(16 + 8d + d^2) + 16 + (16 - 8d + d^2) = 98$</p><p>$48 + 2d^2 = 98$</p><p>$2d^2 = 50$</p><p>$d^2 = 25$</p><p>Отсюда получаем два возможных значения для разности: $d = 5$ и $d = -5$.</p><p>Найдем $a_1$ для каждого случая, используя $a_1 = -4 - d$.</p><p><strong>1.</strong> Если $d = 5$, то $a_1 = -4 - 5 = -9$.</p><p><strong>2.</strong> Если $d = -5$, то $a_1 = -4 - (-5) = 1$.</p><p>Таким образом, существуют две прогрессии, удовлетворяющие заданным условиям.</p><p><strong>Ответ:</strong> $a_1 = -9, d = 5$ или $a_1 = 1, d = -5$.</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" => "1108109" "type" => "task" ] "previous" => array:2 [ "refs" => "1108107" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\BookPage {#1070 #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" => 1097856 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-114" "field_display_title" => "114" "field_folder" => "folder1" "field_image_name" => "114" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Book {#1114} ] #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 {#1108 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Element {#1156 #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" => 1261007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1158 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1158 …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" => "1097857" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097855" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:9 [ 0 => App\Models\Task {#1269 #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" => 1108100 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-376" "field_display_title" => "376" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1270 …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 {#1271 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1272 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1276 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1282 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108100 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-376" "field_display_title" => "376" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1270 …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 {#1271 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1272 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1276 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1282 …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 {#1275 #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" => 1108101 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-377" "field_display_title" => "377" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1274 …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 {#1281 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1283 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1279 …2} "content" => Illuminate\Database\Eloquent\Collection {#1278 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1296 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108101 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-377" "field_display_title" => "377" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1274 …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 {#1281 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1283 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1279 …2} "content" => Illuminate\Database\Eloquent\Collection {#1278 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1296 …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 {#1280 #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" => 1108102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-378" "field_display_title" => "378" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1277 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1297 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1293 …2} "content" => Illuminate\Database\Eloquent\Collection {#1292 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1310 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-378" "field_display_title" => "378" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1277 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1297 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1293 …2} "content" => Illuminate\Database\Eloquent\Collection {#1292 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1310 …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 {#1294 #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" => 1108103 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-379" "field_display_title" => "379" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1291 …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 {#1309 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1311 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1306 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108103 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-379" "field_display_title" => "379" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1291 …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 {#1309 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1311 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1307 …2} "content" => Illuminate\Database\Eloquent\Collection {#1306 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …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 {#1308 #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" => 1108104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-380" "field_display_title" => "380" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1305 …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 {#1323 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1325 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1321 …2} "content" => Illuminate\Database\Eloquent\Collection {#1320 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1338 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-380" "field_display_title" => "380" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1305 …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 {#1323 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1325 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1321 …2} "content" => Illuminate\Database\Eloquent\Collection {#1320 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1338 …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 => "*" ] } 5 => App\Models\Task {#1322 #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" => 1108105 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-381" "field_display_title" => "381" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1319 …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 {#1337 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1339 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1335 …2} "content" => Illuminate\Database\Eloquent\Collection {#1334 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1352 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108105 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-381" "field_display_title" => "381" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1319 …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 {#1337 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1339 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1335 …2} "content" => Illuminate\Database\Eloquent\Collection {#1334 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1352 …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 => "*" ] } 6 => App\Models\Task {#1336 #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" => 1108106 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-382" "field_display_title" => "382" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1333 …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 {#1351 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1353 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1349 …2} "content" => Illuminate\Database\Eloquent\Collection {#1348 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1366 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108106 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-382" "field_display_title" => "382" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1333 …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 {#1351 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1353 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1349 …2} "content" => Illuminate\Database\Eloquent\Collection {#1348 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1366 …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 => "*" ] } 7 => App\Models\Task {#1350 #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" => 1108107 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-383" "field_display_title" => "383" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1347 …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 {#1365 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1367 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1363 …2} "content" => Illuminate\Database\Eloquent\Collection {#1362 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1380 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108107 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-383" "field_display_title" => "383" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1347 …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 {#1365 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1367 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1363 …2} "content" => Illuminate\Database\Eloquent\Collection {#1362 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1380 …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 => "*" ] } 8 => App\Models\Task {#1364 #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" => 1108108 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-384" "field_display_title" => "384" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1361 …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 {#1379 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1381 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1377 …2} "content" => Illuminate\Database\Eloquent\Collection {#1375 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1378 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108108 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-384" "field_display_title" => "384" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1361 …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 {#1379 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1381 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1377 …2} "content" => Illuminate\Database\Eloquent\Collection {#1375 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1378 …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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097856 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-114" "field_display_title" => "114" "field_folder" => "folder1" "field_image_name" => "114" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1073} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1105} "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 {#1108} "content" => Illuminate\Database\Eloquent\Collection {#1157} "next" => array:2 [ "refs" => "1097857" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097855" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1253} ] #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" => 1108108 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-384" "field_display_title" => "384" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "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 {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1104} "content" => Illuminate\Database\Eloquent\Collection {#1101} "next" => array:2 [ "refs" => "1108109" "type" => "task" ] "previous" => array:2 [ "refs" => "1108107" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097} "page" => array:2 [ "refs" => "1097856" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№384 (с. 114)
Условие. №384 (с. 114)
скриншот условия
384. Найдите первый член и разность арифметической прогрессии $a_n$, если:
а) $a_2 + a_4 + a_6 = 18$ и $a_2 \cdot a_4 \cdot a_6 = 120;$
б) $a_1 + a_2 + a_3 = -12$ и $a_1^2 + a_2^2 + a_3^2 = 98.$
Решение. №384 (с. 114)
Решение 2 (rus). №384 (с. 114)
а)
Даны условия для арифметической прогрессии $(a_n)$: $a_2 + a_4 + a_6 = 18$ и $a_2 \cdot a_4 \cdot a_6 = 120$.
Воспользуемся формулой n-го члена арифметической прогрессии $a_n = a_1 + (n-1)d$, где $a_1$ — первый член, а $d$ — разность прогрессии. Выразим члены $a_2, a_4, a_6$ через $a_1$ и $d$:
$a_2 = a_1 + d$
$a_4 = a_1 + 3d$
$a_6 = a_1 + 5d$
Подставим эти выражения в первое уравнение:
$(a_1 + d) + (a_1 + 3d) + (a_1 + 5d) = 18$
$3a_1 + 9d = 18$
Разделив обе части на 3, получим: $a_1 + 3d = 6$.
Левая часть этого уравнения, $a_1 + 3d$, является формулой для $a_4$, следовательно, $a_4 = 6$.
Теперь подставим $a_4 = 6$ во второе уравнение:
$a_2 \cdot 6 \cdot a_6 = 120$, откуда $a_2 \cdot a_6 = 20$.
Выразим $a_2$ и $a_6$ через $a_4$ и $d$:
$a_2 = a_4 - 2d = 6 - 2d$
$a_6 = a_4 + 2d = 6 + 2d$
Подставим эти выражения в уравнение $a_2 \cdot a_6 = 20$:
$(6 - 2d)(6 + 2d) = 20$
$36 - 4d^2 = 20$
$4d^2 = 16$
$d^2 = 4$
Отсюда получаем два возможных значения для разности: $d = 2$ и $d = -2$.
Теперь найдем $a_1$ для каждого значения $d$, используя уравнение $a_1 = 6 - 3d$.
1. Если $d = 2$, то $a_1 = 6 - 3(2) = 0$.
2. Если $d = -2$, то $a_1 = 6 - 3(-2) = 12$.
Таким образом, существуют две прогрессии, удовлетворяющие заданным условиям.
Ответ: $a_1 = 0, d = 2$ или $a_1 = 12, d = -2$.
б)
Даны условия: $a_1 + a_2 + a_3 = -12$ и $a_1^2 + a_2^2 + a_3^2 = 98$.
Выразим члены прогрессии $a_2$ и $a_3$ через $a_1$ и $d$:
$a_2 = a_1 + d$
$a_3 = a_1 + 2d$
Подставим в первое уравнение:
$a_1 + (a_1 + d) + (a_1 + 2d) = -12$
$3a_1 + 3d = -12$
Разделив обе части на 3, получим: $a_1 + d = -4$.
Левая часть этого уравнения, $a_1 + d$, является формулой для $a_2$, следовательно, $a_2 = -4$.
Теперь выразим $a_1$ и $a_3$ через $a_2$ и $d$:
$a_1 = a_2 - d = -4 - d$
$a_3 = a_2 + d = -4 + d$
Подставим эти выражения и значение $a_2 = -4$ во второе уравнение:
$(-4 - d)^2 + (-4)^2 + (-4 + d)^2 = 98$
Раскроем скобки и упростим:
$(16 + 8d + d^2) + 16 + (16 - 8d + d^2) = 98$
$48 + 2d^2 = 98$
$2d^2 = 50$
$d^2 = 25$
Отсюда получаем два возможных значения для разности: $d = 5$ и $d = -5$.
Найдем $a_1$ для каждого случая, используя $a_1 = -4 - d$.
1. Если $d = 5$, то $a_1 = -4 - 5 = -9$.
2. Если $d = -5$, то $a_1 = -4 - (-5) = 1$.
Таким образом, существуют две прогрессии, удовлетворяющие заданным условиям.
Ответ: $a_1 = -9, d = 5$ или $a_1 = 1, d = -5$.
Другие задания:
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