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Номер 318, страница 97 - 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. 13. Числовая последовательность, способы ее задания и свойства - номер 318, страница 97.
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" => 1108036 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-318" "field_display_title" => "318" "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 [ 0 => App\Models\Term {#1042 …30} ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Term {#1044 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:3 [ 0 => App\Models\Term {#1048 …30} 1 => App\Models\Term {#1049 …30} 2 => App\Models\Term {#1050 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 …30} ] #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 [ "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 {#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 [ "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 {#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" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "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 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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" => 1107654 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "13. Числовая последовательность, способы ее задания и свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "93" "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" => 1107654 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "13. Числовая последовательность, способы ее задания и свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145} "field_page_start" => "93" "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 {#1087 #items: array:3 [ 0 => App\Models\Element {#1079 #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" => 1276673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1076 #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 {#1074 #items: array:1 [ 0 => App\Models\Term {#1077 …30} ] #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 {#1075 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 #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 {#1074} "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 {#1075} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1072} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103} "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" => "1108036" "type" => "task" ] "text" => "<p><strong>318.</strong> Исследуйте, какая из последовательностей $(a_n)$, $(b_n)$, $(c_n)$, $(d_n)$, $(e_n)$, $(f_n)$ является: а) возрастающей; б) убывающей:</p><p><strong>$(a_n)$:</strong> $\frac{1}{2}; \frac{2}{3}; \frac{3}{4}; \frac{4}{5}; \frac{5}{6}$</p><p><strong>$(b_n)$:</strong> $-4; -16; -64; -256.$</p><p><strong>$(c_n)$:</strong> $-5; 25; -125; 625.$</p><p><strong>$(d_n)$:</strong> $\frac{3}{10}; \frac{9}{100}; \frac{27}{1000}; \frac{81}{10000}$</p><p><strong>$(e_n)$:</strong> $5; 2; 5; 5; 2; 5; 5; 5; 2.$</p><p><strong>$(f_n)$:</strong> $0,1; 0,11; 0,111; 0,1111.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "318-1.jpg" "alt" => null "width" => "1575" "height" => 490 "path" => "/media/algebra_09/soltan-u19/0-1/318-1.webp?ts=1752834354" ] ] ] #original: array:7 [ "id" => 1276673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1108036" "type" => "task" ] "text" => "<p><strong>318.</strong> Исследуйте, какая из последовательностей $(a_n)$, $(b_n)$, $(c_n)$, $(d_n)$, $(e_n)$, $(f_n)$ является: а) возрастающей; б) убывающей:</p><p><strong>$(a_n)$:</strong> $\frac{1}{2}; \frac{2}{3}; \frac{3}{4}; \frac{4}{5}; \frac{5}{6}$</p><p><strong>$(b_n)$:</strong> $-4; -16; -64; -256.$</p><p><strong>$(c_n)$:</strong> $-5; 25; -125; 625.$</p><p><strong>$(d_n)$:</strong> $\frac{3}{10}; \frac{9}{100}; \frac{27}{1000}; \frac{81}{10000}$</p><p><strong>$(e_n)$:</strong> $5; 2; 5; 5; 2; 5; 5; 5; 2.$</p><p><strong>$(f_n)$:</strong> $0,1; 0,11; 0,111; 0,1111.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "318-1.jpg" "alt" => null "width" => "1575" "height" => 490 "path" => "/media/algebra_09/soltan-u19/0-1/318-1.webp?ts=1752834354" ] ] ] #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 {#1102 #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" => 1277964 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Edition {#1111 #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 {#1108 #items: array:1 [ 0 => App\Models\Term {#1105 …30} ] #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 {#1106 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1149 #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 {#1108} "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 {#1106} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1148} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1149} "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" => "1108036" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "318-1.jpg" "alt" => null "width" => "675" "height" => 434 "path" => "/media/algebra_09/soltan-u19/1-1/318-1.webp?ts=1752831055" ] ] ] #original: array:6 [ "id" => 1277964 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1152} "task" => array:2 [ "refs" => "1108036" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "318-1.jpg" "alt" => null "width" => "675" "height" => 434 "path" => "/media/algebra_09/soltan-u19/1-1/318-1.webp?ts=1752831055" ] ] ] #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 {#1150 #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" => 1553919 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Edition {#1151 #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 {#1154 #items: array:1 [ 0 => App\Models\Term {#1153 …30} ] #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 {#1155 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1157 #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 {#1154} "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 {#1155} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1157} "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" => "1108036" "type" => "task" ] "text" => "<p><strong>(a$_n$):</strong> Последовательность задана членами $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \dots$.</p><p>Общий член этой последовательности можно записать формулой $a_n = \frac{n}{n+1}$.</p><p>Для того чтобы исследовать последовательность на монотонность, сравним два соседних члена: $a_n$ и $a_{n+1}$. Член $a_{n+1}$ имеет вид $a_{n+1} = \frac{n+1}{(n+1)+1} = \frac{n+1}{n+2}$.</p><p>Рассмотрим разность $a_{n+1} - a_n$:</p><p>$a_{n+1} - a_n = \frac{n+1}{n+2} - \frac{n}{n+1} = \frac{(n+1)^2 - n(n+2)}{(n+2)(n+1)} = \frac{n^2 + 2n + 1 - n^2 - 2n}{(n+1)(n+2)} = \frac{1}{(n+1)(n+2)}$.</p><p>Так как $n \ge 1$, то знаменатель $(n+1)(n+2)$ всегда положителен. Следовательно, разность $a_{n+1} - a_n$ всегда больше нуля. Это означает, что $a_{n+1} > a_n$ для любого натурального $n$. Таким образом, последовательность является возрастающей.</p><p>Альтернативный способ: представим $a_n = \frac{n}{n+1} = \frac{n+1-1}{n+1} = 1 - \frac{1}{n+1}$. С увеличением $n$ знаменатель $n+1$ растет, дробь $\frac{1}{n+1}$ уменьшается, а значит, разность $1 - \frac{1}{n+1}$ увеличивается. Следовательно, последовательность возрастающая.</p><p><strong>Ответ:</strong> последовательность $(a_n)$ является возрастающей.</p><p><strong>(b$_n$):</strong> Последовательность задана членами $-4, -16, -64, -256, \dots$.</p><p>Общий член этой последовательности можно записать формулой $b_n = -4^n$.</p><p>Сравним соседние члены $b_n$ и $b_{n+1}$. Член $b_{n+1}$ имеет вид $b_{n+1} = -4^{n+1}$.</p><p>Рассмотрим разность $b_{n+1} - b_n$: $b_{n+1} - b_n = -4^{n+1} - (-4^n) = -4^{n+1} + 4^n = 4^n(-4 + 1) = -3 \cdot 4^n$.</p><p>Так как $4^n > 0$ для любого натурального $n$, то произведение $-3 \cdot 4^n$ всегда отрицательно. Следовательно, разность $b_{n+1} - b_n < 0$, что означает $b_{n+1} < b_n$ для любого $n$. Таким образом, последовательность является убывающей.</p><p><strong>Ответ:</strong> последовательность $(b_n)$ является убывающей.</p><p><strong>(c$_n$):</strong> Последовательность задана членами $-5, 25, -125, 625, \dots$.</p><p>Общий член этой последовательности можно записать формулой $c_n = (-5)^n$.</p><p>Это знакочередующаяся последовательность. Сравним первые несколько членов:</p><p>$c_1 = -5$, $c_2 = 25$, $c_3 = -125$.</p><p>Видим, что $c_2 > c_1$ ($25 > -5$), но $c_3 < c_2$ ($-125 < 25$).</p><p>Поскольку условие $c_{n+1} > c_n$ (для возрастающей) и условие $c_{n+1} < c_n$ (для убывающей) не выполняются для всех $n$, последовательность не является ни возрастающей, ни убывающей.</p><p><strong>Ответ:</strong> последовательность $(c_n)$ не является ни возрастающей, ни убывающей.</p><p><strong>(d$_n$):</strong> Последовательность задана членами $\frac{3}{10}, \frac{9}{100}, \frac{27}{1000}, \frac{81}{10000}, \dots$.</p><p>Общий член этой последовательности можно записать формулой $d_n = \frac{3^n}{10^n} = \left(\frac{3}{10}\right)^n$.</p><p>Это геометрическая прогрессия с первым членом $d_1 = \frac{3}{10}$ и знаменателем $q = \frac{3}{10}$.</p><p>Так как все члены последовательности положительны, мы можем рассмотреть их отношение: $\frac{d_{n+1}}{d_n} = \frac{(3/10)^{n+1}}{(3/10)^n} = \frac{3}{10}$.</p><p>Поскольку $\frac{d_{n+1}}{d_n} = \frac{3}{10} < 1$, то $d_{n+1} < d_n$ для любого натурального $n$. Следовательно, последовательность является убывающей.</p><p><strong>Ответ:</strong> последовательность $(d_n)$ является убывающей.</p><p><strong>(e$_n$):</strong> Последовательность задана конечным набором членов $5, 2, 5, 5, 2, 5, 5, 5, 2$.</p><p>Сравним первые несколько членов:</p><p>$e_1 = 5$, $e_2 = 2$. Здесь $e_2 < e_1$.</p><p>$e_2 = 2$, $e_3 = 5$. Здесь $e_3 > e_2$.</p><p>Так как в последовательности есть как участки убывания ($e_2 < e_1$), так и участки возрастания ($e_3 > e_2$), она не является монотонной.</p><p><strong>Ответ:</strong> последовательность $(e_n)$ не является ни возрастающей, ни убывающей.</p><p><strong>(f$_n$):</strong> Последовательность задана членами $0.1, 0.11, 0.111, 0.1111, \dots$.</p><p>Запишем члены последовательности: $f_1 = 0.1$, $f_2 = 0.11$, $f_3 = 0.111$, и так далее.</p><p>Сравним два соседних члена $f_n$ и $f_{n+1}$. Каждый следующий член получается из предыдущего добавлением цифры 1 в следующем разряде после запятой.</p><p>Например, $f_2 = 0.11 = 0.1 + 0.01 = f_1 + 0.01$. Очевидно, $f_2 > f_1$.</p><p>В общем виде, разность $f_{n+1} - f_n = 0.\underbrace{00\dots0}_{n}1 = 10^{-(n+1)}$.</p><p>Так как разность $f_{n+1} - f_n$ всегда положительна ($10^{-(n+1)} > 0$ для любого $n$), то $f_{n+1} > f_n$ для всех $n$. Следовательно, последовательность является возрастающей.</p><p><strong>Ответ:</strong> последовательность $(f_n)$ является возрастающей.</p><p>Таким образом, подводя итог исследования:</p><p><strong>а) возрастающей</strong> являются последовательности: $(a_n)$ и $(f_n)$.</p><p><strong>б) убывающей</strong> являются последовательности: $(b_n)$ и $(d_n)$.</p><p>Последовательности $(c_n)$ и $(e_n)$ не являются ни возрастающими, ни убывающими.</p>" ] #original: array:6 [ "id" => 1553919 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160} "task" => array:2 [ "refs" => "1108036" "type" => "task" ] "text" => "<p><strong>(a$_n$):</strong> Последовательность задана членами $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \dots$.</p><p>Общий член этой последовательности можно записать формулой $a_n = \frac{n}{n+1}$.</p><p>Для того чтобы исследовать последовательность на монотонность, сравним два соседних члена: $a_n$ и $a_{n+1}$. Член $a_{n+1}$ имеет вид $a_{n+1} = \frac{n+1}{(n+1)+1} = \frac{n+1}{n+2}$.</p><p>Рассмотрим разность $a_{n+1} - a_n$:</p><p>$a_{n+1} - a_n = \frac{n+1}{n+2} - \frac{n}{n+1} = \frac{(n+1)^2 - n(n+2)}{(n+2)(n+1)} = \frac{n^2 + 2n + 1 - n^2 - 2n}{(n+1)(n+2)} = \frac{1}{(n+1)(n+2)}$.</p><p>Так как $n \ge 1$, то знаменатель $(n+1)(n+2)$ всегда положителен. Следовательно, разность $a_{n+1} - a_n$ всегда больше нуля. Это означает, что $a_{n+1} > a_n$ для любого натурального $n$. Таким образом, последовательность является возрастающей.</p><p>Альтернативный способ: представим $a_n = \frac{n}{n+1} = \frac{n+1-1}{n+1} = 1 - \frac{1}{n+1}$. С увеличением $n$ знаменатель $n+1$ растет, дробь $\frac{1}{n+1}$ уменьшается, а значит, разность $1 - \frac{1}{n+1}$ увеличивается. Следовательно, последовательность возрастающая.</p><p><strong>Ответ:</strong> последовательность $(a_n)$ является возрастающей.</p><p><strong>(b$_n$):</strong> Последовательность задана членами $-4, -16, -64, -256, \dots$.</p><p>Общий член этой последовательности можно записать формулой $b_n = -4^n$.</p><p>Сравним соседние члены $b_n$ и $b_{n+1}$. Член $b_{n+1}$ имеет вид $b_{n+1} = -4^{n+1}$.</p><p>Рассмотрим разность $b_{n+1} - b_n$: $b_{n+1} - b_n = -4^{n+1} - (-4^n) = -4^{n+1} + 4^n = 4^n(-4 + 1) = -3 \cdot 4^n$.</p><p>Так как $4^n > 0$ для любого натурального $n$, то произведение $-3 \cdot 4^n$ всегда отрицательно. Следовательно, разность $b_{n+1} - b_n < 0$, что означает $b_{n+1} < b_n$ для любого $n$. Таким образом, последовательность является убывающей.</p><p><strong>Ответ:</strong> последовательность $(b_n)$ является убывающей.</p><p><strong>(c$_n$):</strong> Последовательность задана членами $-5, 25, -125, 625, \dots$.</p><p>Общий член этой последовательности можно записать формулой $c_n = (-5)^n$.</p><p>Это знакочередующаяся последовательность. Сравним первые несколько членов:</p><p>$c_1 = -5$, $c_2 = 25$, $c_3 = -125$.</p><p>Видим, что $c_2 > c_1$ ($25 > -5$), но $c_3 < c_2$ ($-125 < 25$).</p><p>Поскольку условие $c_{n+1} > c_n$ (для возрастающей) и условие $c_{n+1} < c_n$ (для убывающей) не выполняются для всех $n$, последовательность не является ни возрастающей, ни убывающей.</p><p><strong>Ответ:</strong> последовательность $(c_n)$ не является ни возрастающей, ни убывающей.</p><p><strong>(d$_n$):</strong> Последовательность задана членами $\frac{3}{10}, \frac{9}{100}, \frac{27}{1000}, \frac{81}{10000}, \dots$.</p><p>Общий член этой последовательности можно записать формулой $d_n = \frac{3^n}{10^n} = \left(\frac{3}{10}\right)^n$.</p><p>Это геометрическая прогрессия с первым членом $d_1 = \frac{3}{10}$ и знаменателем $q = \frac{3}{10}$.</p><p>Так как все члены последовательности положительны, мы можем рассмотреть их отношение: $\frac{d_{n+1}}{d_n} = \frac{(3/10)^{n+1}}{(3/10)^n} = \frac{3}{10}$.</p><p>Поскольку $\frac{d_{n+1}}{d_n} = \frac{3}{10} < 1$, то $d_{n+1} < d_n$ для любого натурального $n$. Следовательно, последовательность является убывающей.</p><p><strong>Ответ:</strong> последовательность $(d_n)$ является убывающей.</p><p><strong>(e$_n$):</strong> Последовательность задана конечным набором членов $5, 2, 5, 5, 2, 5, 5, 5, 2$.</p><p>Сравним первые несколько членов:</p><p>$e_1 = 5$, $e_2 = 2$. Здесь $e_2 < e_1$.</p><p>$e_2 = 2$, $e_3 = 5$. Здесь $e_3 > e_2$.</p><p>Так как в последовательности есть как участки убывания ($e_2 < e_1$), так и участки возрастания ($e_3 > e_2$), она не является монотонной.</p><p><strong>Ответ:</strong> последовательность $(e_n)$ не является ни возрастающей, ни убывающей.</p><p><strong>(f$_n$):</strong> Последовательность задана членами $0.1, 0.11, 0.111, 0.1111, \dots$.</p><p>Запишем члены последовательности: $f_1 = 0.1$, $f_2 = 0.11$, $f_3 = 0.111$, и так далее.</p><p>Сравним два соседних члена $f_n$ и $f_{n+1}$. Каждый следующий член получается из предыдущего добавлением цифры 1 в следующем разряде после запятой.</p><p>Например, $f_2 = 0.11 = 0.1 + 0.01 = f_1 + 0.01$. Очевидно, $f_2 > f_1$.</p><p>В общем виде, разность $f_{n+1} - f_n = 0.\underbrace{00\dots0}_{n}1 = 10^{-(n+1)}$.</p><p>Так как разность $f_{n+1} - f_n$ всегда положительна ($10^{-(n+1)} > 0$ для любого $n$), то $f_{n+1} > f_n$ для всех $n$. Следовательно, последовательность является возрастающей.</p><p><strong>Ответ:</strong> последовательность $(f_n)$ является возрастающей.</p><p>Таким образом, подводя итог исследования:</p><p><strong>а) возрастающей</strong> являются последовательности: $(a_n)$ и $(f_n)$.</p><p><strong>б) убывающей</strong> являются последовательности: $(b_n)$ и $(d_n)$.</p><p>Последовательности $(c_n)$ и $(e_n)$ не являются ни возрастающими, ни убывающими.</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" => "1108037" "type" => "task" ] "previous" => array:2 [ "refs" => "1108035" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1094 #items: array:1 [ 0 => App\Models\BookPage {#1092 #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" => 1097839 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-97" "field_display_title" => "97" "field_folder" => "folder1" "field_image_name" => "97" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1109 #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 {#1101 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Element {#1069 #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" => 1260990 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Edition {#1098 …30} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1097839" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "97-1.jpg" "alt" => null "width" => "2009" "height" => 2972 "path" => "/media/algebra_09/soltan-u19/0-folder1/97-1.webp?ts=1752141137" ] ] ] #original: array:6 [ "id" => 1260990 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099} "book_page" => array:2 [ "refs" => "1097839" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "97-1.jpg" "alt" => null "width" => "2009" "height" => 2972 "path" => "/media/algebra_09/soltan-u19/0-folder1/97-1.webp?ts=1752141137" ] ] ] #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" => "1097840" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097838" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1656 #items: array:6 [ 0 => App\Models\Task {#1668 #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" => 1108035 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => "97" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-317" "field_display_title" => "317" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1669 #items: array:1 [ …1] #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 {#1670 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1671 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1672 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1698 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108036" "type" => "task" ] "previous" => array:2 [ "refs" => "1108034" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1699 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097838" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108035 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => "97" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-317" "field_display_title" => "317" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1669} "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 {#1670} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1671} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1672} "content" => Illuminate\Database\Eloquent\Collection {#1698} "next" => array:2 [ "refs" => "1108036" "type" => "task" ] "previous" => array:2 [ "refs" => "1108034" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1699} "page" => array:2 [ "refs" => "1097838" "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 => "*" ] } 1 => App\Models\Task {#1697 #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" => 1108036 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-318" "field_display_title" => "318" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1700 #items: array:1 [ …1] #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 {#1701 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1702 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1703 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1704 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108037" "type" => "task" ] "previous" => array:2 [ "refs" => "1108035" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1705 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097839" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108036 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-318" "field_display_title" => "318" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1700} "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 {#1701} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1702} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1703} "content" => Illuminate\Database\Eloquent\Collection {#1704} "next" => array:2 [ "refs" => "1108037" "type" => "task" ] "previous" => array:2 [ "refs" => "1108035" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1705} "page" => array:2 [ "refs" => "1097839" "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 => "*" ] } 2 => App\Models\Task {#1706 #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" => 1108037 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-319" "field_display_title" => "319" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1707 #items: array:1 [ …1] #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 {#1708 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1709 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1710 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1728 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108038" "type" => "task" ] "previous" => array:2 [ "refs" => "1108036" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1735 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097839" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108037 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-319" "field_display_title" => "319" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1707} "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 {#1708} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1709} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1710} "content" => Illuminate\Database\Eloquent\Collection {#1728} "next" => array:2 [ "refs" => "1108038" "type" => "task" ] "previous" => array:2 [ "refs" => "1108036" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1735} "page" => array:2 [ "refs" => "1097839" "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 => "*" ] } 3 => App\Models\Task {#1719 #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" => 1108038 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-320" "field_display_title" => "320" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1720 #items: array:1 [ …1] #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 {#1721 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1726 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1725 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1733 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108039" "type" => "task" ] "previous" => array:2 [ "refs" => "1108037" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1749 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097839" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108038 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-320" "field_display_title" => "320" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1720} "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 {#1721} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1726} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1725} "content" => Illuminate\Database\Eloquent\Collection {#1733} "next" => array:2 [ "refs" => "1108039" "type" => "task" ] "previous" => array:2 [ "refs" => "1108037" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1749} "page" => array:2 [ "refs" => "1097839" "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 => "*" ] } 4 => App\Models\Task {#1716 #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" => 1108039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-321" "field_display_title" => "321" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1714 #items: array:1 [ …1] #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 {#1715 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1732 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1736 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1747 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108040" "type" => "task" ] "previous" => array:2 [ "refs" => "1108038" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1763 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097839" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-321" "field_display_title" => "321" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1714} "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 {#1715} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1732} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1736} "content" => Illuminate\Database\Eloquent\Collection {#1747} "next" => array:2 [ "refs" => "1108040" "type" => "task" ] "previous" => array:2 [ "refs" => "1108038" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1763} "page" => array:2 [ "refs" => "1097839" "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 => "*" ] } 5 => App\Models\Task {#1712 #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" => 1108040 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-322" "field_display_title" => "322" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1711 #items: array:1 [ …1] #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 {#1713 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1746 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1750 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1761 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108041" "type" => "task" ] "previous" => array:2 [ "refs" => "1108039" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1777 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097839" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108040 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-322" "field_display_title" => "322" "field_outside_task" => null "field_task_type" => 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[] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097839 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-97" "field_display_title" => "97" "field_folder" => "folder1" "field_image_name" => "97" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1109} "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 {#1101} "content" => Illuminate\Database\Eloquent\Collection {#1084} "next" => array:2 [ "refs" => "1097840" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097838" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1656} ] #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" => 1108036 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-318" "field_display_title" => "318" "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 {#1087} "next" => array:2 [ "refs" => "1108037" "type" => "task" ] "previous" => array:2 [ "refs" => "1108035" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1082} "page" => array:2 [ "refs" => "1097839" "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 => "*" ] }
№318 (с. 97)
Условие. №318 (с. 97)
скриншот условия
318. Исследуйте, какая из последовательностей $(a_n)$, $(b_n)$, $(c_n)$, $(d_n)$, $(e_n)$, $(f_n)$ является: а) возрастающей; б) убывающей:
$(a_n)$: $\frac{1}{2}; \frac{2}{3}; \frac{3}{4}; \frac{4}{5}; \frac{5}{6}$
$(b_n)$: $-4; -16; -64; -256.$
$(c_n)$: $-5; 25; -125; 625.$
$(d_n)$: $\frac{3}{10}; \frac{9}{100}; \frac{27}{1000}; \frac{81}{10000}$
$(e_n)$: $5; 2; 5; 5; 2; 5; 5; 5; 2.$
$(f_n)$: $0,1; 0,11; 0,111; 0,1111.$
Решение. №318 (с. 97)
Решение 2 (rus). №318 (с. 97)
(a$_n$): Последовательность задана членами $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \dots$.
Общий член этой последовательности можно записать формулой $a_n = \frac{n}{n+1}$.
Для того чтобы исследовать последовательность на монотонность, сравним два соседних члена: $a_n$ и $a_{n+1}$. Член $a_{n+1}$ имеет вид $a_{n+1} = \frac{n+1}{(n+1)+1} = \frac{n+1}{n+2}$.
Рассмотрим разность $a_{n+1} - a_n$:
$a_{n+1} - a_n = \frac{n+1}{n+2} - \frac{n}{n+1} = \frac{(n+1)^2 - n(n+2)}{(n+2)(n+1)} = \frac{n^2 + 2n + 1 - n^2 - 2n}{(n+1)(n+2)} = \frac{1}{(n+1)(n+2)}$.
Так как $n \ge 1$, то знаменатель $(n+1)(n+2)$ всегда положителен. Следовательно, разность $a_{n+1} - a_n$ всегда больше нуля. Это означает, что $a_{n+1} > a_n$ для любого натурального $n$. Таким образом, последовательность является возрастающей.
Альтернативный способ: представим $a_n = \frac{n}{n+1} = \frac{n+1-1}{n+1} = 1 - \frac{1}{n+1}$. С увеличением $n$ знаменатель $n+1$ растет, дробь $\frac{1}{n+1}$ уменьшается, а значит, разность $1 - \frac{1}{n+1}$ увеличивается. Следовательно, последовательность возрастающая.
Ответ: последовательность $(a_n)$ является возрастающей.
(b$_n$): Последовательность задана членами $-4, -16, -64, -256, \dots$.
Общий член этой последовательности можно записать формулой $b_n = -4^n$.
Сравним соседние члены $b_n$ и $b_{n+1}$. Член $b_{n+1}$ имеет вид $b_{n+1} = -4^{n+1}$.
Рассмотрим разность $b_{n+1} - b_n$: $b_{n+1} - b_n = -4^{n+1} - (-4^n) = -4^{n+1} + 4^n = 4^n(-4 + 1) = -3 \cdot 4^n$.
Так как $4^n > 0$ для любого натурального $n$, то произведение $-3 \cdot 4^n$ всегда отрицательно. Следовательно, разность $b_{n+1} - b_n < 0$, что означает $b_{n+1} < b_n$ для любого $n$. Таким образом, последовательность является убывающей.
Ответ: последовательность $(b_n)$ является убывающей.
(c$_n$): Последовательность задана членами $-5, 25, -125, 625, \dots$.
Общий член этой последовательности можно записать формулой $c_n = (-5)^n$.
Это знакочередующаяся последовательность. Сравним первые несколько членов:
$c_1 = -5$, $c_2 = 25$, $c_3 = -125$.
Видим, что $c_2 > c_1$ ($25 > -5$), но $c_3 < c_2$ ($-125 < 25$).
Поскольку условие $c_{n+1} > c_n$ (для возрастающей) и условие $c_{n+1} < c_n$ (для убывающей) не выполняются для всех $n$, последовательность не является ни возрастающей, ни убывающей.
Ответ: последовательность $(c_n)$ не является ни возрастающей, ни убывающей.
(d$_n$): Последовательность задана членами $\frac{3}{10}, \frac{9}{100}, \frac{27}{1000}, \frac{81}{10000}, \dots$.
Общий член этой последовательности можно записать формулой $d_n = \frac{3^n}{10^n} = \left(\frac{3}{10}\right)^n$.
Это геометрическая прогрессия с первым членом $d_1 = \frac{3}{10}$ и знаменателем $q = \frac{3}{10}$.
Так как все члены последовательности положительны, мы можем рассмотреть их отношение: $\frac{d_{n+1}}{d_n} = \frac{(3/10)^{n+1}}{(3/10)^n} = \frac{3}{10}$.
Поскольку $\frac{d_{n+1}}{d_n} = \frac{3}{10} < 1$, то $d_{n+1} < d_n$ для любого натурального $n$. Следовательно, последовательность является убывающей.
Ответ: последовательность $(d_n)$ является убывающей.
(e$_n$): Последовательность задана конечным набором членов $5, 2, 5, 5, 2, 5, 5, 5, 2$.
Сравним первые несколько членов:
$e_1 = 5$, $e_2 = 2$. Здесь $e_2 < e_1$.
$e_2 = 2$, $e_3 = 5$. Здесь $e_3 > e_2$.
Так как в последовательности есть как участки убывания ($e_2 < e_1$), так и участки возрастания ($e_3 > e_2$), она не является монотонной.
Ответ: последовательность $(e_n)$ не является ни возрастающей, ни убывающей.
(f$_n$): Последовательность задана членами $0.1, 0.11, 0.111, 0.1111, \dots$.
Запишем члены последовательности: $f_1 = 0.1$, $f_2 = 0.11$, $f_3 = 0.111$, и так далее.
Сравним два соседних члена $f_n$ и $f_{n+1}$. Каждый следующий член получается из предыдущего добавлением цифры 1 в следующем разряде после запятой.
Например, $f_2 = 0.11 = 0.1 + 0.01 = f_1 + 0.01$. Очевидно, $f_2 > f_1$.
В общем виде, разность $f_{n+1} - f_n = 0.\underbrace{00\dots0}_{n}1 = 10^{-(n+1)}$.
Так как разность $f_{n+1} - f_n$ всегда положительна ($10^{-(n+1)} > 0$ для любого $n$), то $f_{n+1} > f_n$ для всех $n$. Следовательно, последовательность является возрастающей.
Ответ: последовательность $(f_n)$ является возрастающей.
Таким образом, подводя итог исследования:
а) возрастающей являются последовательности: $(a_n)$ и $(f_n)$.
б) убывающей являются последовательности: $(b_n)$ и $(d_n)$.
Последовательности $(c_n)$ и $(e_n)$ не являются ни возрастающими, ни убывающими.
Другие задания:
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