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Номер 317, страница 96 - 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 317, страница 96.
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" => 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 {#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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#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 {#1166 #items: array:2 [ 0 => App\Models\Branch {#1069 #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 {#1072 #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 {#1070 #items: array:1 [ 0 => App\Models\Book {#1073 #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 {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101 #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 {#1072} "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 {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #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 {#1102 #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 {#1103 #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 {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #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 {#1135 …2} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1165 …2} ] #original: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1165 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 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 {#1103} "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 {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #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 {#1169 #items: array:3 [ 0 => App\Models\Element {#1178 #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" => 1276672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Edition {#1179 #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 {#1181 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108035" "type" => "task" ] "text" => "<p><strong>317.</strong> Установите, на каком из рисунков 32–35 изображен график:</p><p>а) возрастающей последовательности; б) убывающей последовательности.</p><p><strong>Рисунок 32</strong></p><p><svg viewBox="0 0 120 160" xmlns="http://www.w3.org/2000/svg" width="120" height="160"><!-- Grid lines --><!-- Horizontal grid lines --><line x1="20" y1="10" x2="100" y2="10" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="20" x2="100" y2="20" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="30" x2="100" y2="30" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="40" x2="100" y2="40" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="50" x2="100" y2="50" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="60" x2="100" y2="60" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="70" x2="100" y2="70" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="80" x2="100" y2="80" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="90" x2="100" y2="90" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="100" x2="100" y2="100" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="110" x2="100" y2="110" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="120" x2="100" y2="120" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="130" x2="100" y2="130" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="140" x2="100" y2="140" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="150" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Vertical grid lines --><line x1="20" y1="10" x2="20" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="30" y1="10" x2="30" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="40" y1="10" x2="40" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="50" y1="10" x2="50" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="60" y1="10" x2="60" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="70" y1="10" x2="70" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="80" y1="10" x2="80" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="90" y1="10" x2="90" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="100" y1="10" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Axes (Origin at 20, 100) --><line x1="20" y1="100" x2="100" y2="100" stroke="black" stroke-width="1.5"/><!-- X-axis --><line x1="20" y1="20" x2="20" y2="140" stroke="black" stroke-width="1.5"/><!-- Y-axis --><!-- Axis labels --><text x="15" y="105" font-family="sans-serif" font-size="8px">O</text><text x="100" y="105" font-family="sans-serif" font-size="8px">n</text><text x="10" y="25" font-family="sans-serif" font-size="8px">a<tspan baseline-shift="sub" font-size="6px">n</tspan></text><text x="10" y="85" font-family="sans-serif" font-size="8px">1-</text><!-- Y-label for 1. At y=100 - 1*20 = 80px. Text baseline is 80+5=85. --><text x="57" y="110" font-family="sans-serif" font-size="8px">4</text><text x="77" y="110" font-family="sans-serif" font-size="8px">6</text><!-- Formula (placed visually like in image) --><text x="50" y="40" font-family="sans-serif" font-size="8px">$a_n = \frac{n-3}{2}$</text><!-- Plotted points (OX=20, OY=100, unit_x_px=10, unit_y_px=20) --><circle cx="30" cy="120" r="2" fill="black"/><!-- n=1, a=-1: (20+1*10, 100 - (-1)*20) = (30, 120) --><circle cx="40" cy="110" r="2" fill="black"/><!-- n=2, a=-0.5: (20+2*10, 100 - (-0.5)*20) = (40, 110) --><circle cx="50" cy="100" r="2" fill="black"/><!-- n=3, a=0: (20+3*10, 100 - 0*20) = (50, 100) --><circle cx="60" cy="90" r="2" fill="black"/><!-- n=4, a=0.5: (20+4*10, 100 - 0.5*20) = (60, 90) --><circle cx="70" cy="80" r="2" fill="black"/><!-- n=5, a=1: (20+5*10, 100 - 1*20) = (70, 80) --><circle cx="80" cy="70" r="2" fill="black"/><!-- n=6, a=1.5: (20+6*10, 100 - 1.5*20) = (80, 70) --></svg></p><p>$a_n = \frac{n-3}{2}$</p><p><strong>Рисунок 33</strong></p><p><svg viewBox="0 0 120 160" xmlns="http://www.w3.org/2000/svg" width="120" height="160"><!-- Grid lines --><!-- Horizontal grid lines --><line x1="20" y1="10" x2="100" y2="10" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="20" x2="100" y2="20" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="30" x2="100" y2="30" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="40" x2="100" y2="40" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="50" x2="100" y2="50" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="60" x2="100" y2="60" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="70" x2="100" y2="70" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="80" x2="100" y2="80" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="90" x2="100" y2="90" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="100" x2="100" y2="100" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="110" x2="100" y2="110" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="120" x2="100" y2="120" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="130" x2="100" y2="130" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="140" x2="100" y2="140" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="150" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Vertical grid lines --><line x1="20" y1="10" x2="20" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="30" y1="10" x2="30" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="40" y1="10" x2="40" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="50" y1="10" x2="50" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="60" y1="10" x2="60" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="70" y1="10" x2="70" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="80" y1="10" x2="80" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="90" y1="10" x2="90" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="100" y1="10" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Axes (Origin at 20, 100) --><line x1="20" y1="100" x2="100" y2="100" stroke="black" stroke-width="1.5"/><!-- X-axis --><line x1="20" y1="20" x2="20" y2="140" stroke="black" stroke-width="1.5"/><!-- Y-axis --><!-- Axis labels --><text x="15" y="105" font-family="sans-serif" font-size="8px">O</text><text x="100" y="105" font-family="sans-serif" font-size="8px">n</text><text x="10" y="25" font-family="sans-serif" font-size="8px">b<tspan baseline-shift="sub" font-size="6px">n</tspan></text><text x="10" y="85" font-family="sans-serif" font-size="8px">1-</text><!-- Y-label for 1 --><text x="77" y="110" font-family="sans-serif" font-size="8px">6</text><!-- Formula --><text x="50" y="40" font-family="sans-serif" font-size="8px">$b_n = \frac{4}{n} - 2$</text><!-- Plotted points (OX=20, OY=100, unit_x_px=10, unit_y_px=20) --><circle cx="30" cy="60" r="2" fill="black"/><!-- n=1, b=2: (20+1*10, 100 - 2*20) = (30, 60) --><circle cx="40" cy="100" r="2" fill="black"/><!-- n=2, b=0: (20+2*10, 100 - 0*20) = (40, 100) --><circle cx="50" cy="113.33" r="2" fill="black"/><!-- n=3, b=-2/3: (20+3*10, 100 - (-2/3)*20) = (50, 113.33) --><circle cx="60" cy="120" r="2" fill="black"/><!-- n=4, b=-1: (20+4*10, 100 - (-1)*20) = (60, 120) --><circle cx="70" cy="124" r="2" fill="black"/><!-- n=5, b=-1.2: (20+5*10, 100 - (-1.2)*20) = (70, 124) --><circle cx="80" cy="126.67" r="2" fill="black"/><!-- n=6, b=-4/3: (20+6*10, 100 - (-4/3)*20) = (80, 126.67) --></svg></p><p>$b_n = \frac{4}{n} - 2$</p><p><strong>Рисунок 34</strong></p><p><svg viewBox="0 0 120 160" xmlns="http://www.w3.org/2000/svg" width="120" height="160"><!-- Grid lines --><!-- Horizontal grid lines --><line x1="20" y1="10" x2="100" y2="10" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="20" x2="100" y2="20" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="30" x2="100" y2="30" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="40" x2="100" y2="40" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="50" x2="100" y2="50" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="60" x2="100" y2="60" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="70" x2="100" y2="70" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="80" x2="100" y2="80" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="90" x2="100" y2="90" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="100" x2="100" y2="100" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="110" x2="100" y2="110" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="120" x2="100" y2="120" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="130" x2="100" y2="130" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="140" x2="100" y2="140" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="150" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Vertical grid lines --><line x1="20" y1="10" x2="20" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="30" y1="10" x2="30" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="40" y1="10" x2="40" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="50" y1="10" x2="50" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="60" y1="10" x2="60" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="70" y1="10" x2="70" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="80" y1="10" x2="80" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="90" y1="10" x2="90" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="100" y1="10" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Axes (Origin at 20, 100) --><line x1="20" y1="100" x2="100" y2="100" stroke="black" stroke-width="1.5"/><!-- X-axis --><line x1="20" y1="20" x2="20" y2="150" stroke="black" stroke-width="1.5"/><!-- Y-axis --><!-- Axis labels --><text x="15" y="105" font-family="sans-serif" font-size="8px">O</text><text x="100" y="105" font-family="sans-serif" font-size="8px">n</text><text x="10" y="25" font-family="sans-serif" font-size="8px">c<tspan baseline-shift="sub" font-size="6px">n</tspan></text><text x="10" y="85" font-family="sans-serif" font-size="8px">1-</text><!-- Y-label for 1 --><text x="37" y="110" font-family="sans-serif" font-size="8px">2</text><text x="57" y="110" font-family="sans-serif" font-size="8px">4</text><text x="77" y="110" font-family="sans-serif" font-size="8px">6</text><!-- Formula --><text x="50" y="40" font-family="sans-serif" font-size="8px">$c_n = (-1)^n \cdot \frac{n}{2}$</text><!-- Plotted points (OX=20, OY=100, unit_x_px=10, unit_y_px=20) --><circle cx="30" cy="110" r="2" fill="black"/><!-- n=1, c=-0.5: (20+1*10, 100 - (-0.5)*20) = (30, 110) --><circle cx="40" cy="80" r="2" fill="black"/><!-- n=2, c=1: (20+2*10, 100 - 1*20) = (40, 80) --><circle cx="50" cy="130" r="2" fill="black"/><!-- n=3, c=-1.5: (20+3*10, 100 - (-1.5)*20) = (50, 130) --><circle cx="60" cy="60" r="2" fill="black"/><!-- n=4, c=2: (20+4*10, 100 - 2*20) = (60, 60) --><circle cx="70" cy="150" r="2" fill="black"/><!-- n=5, c=-2.5: (20+5*10, 100 - (-2.5)*20) = (70, 150) --><circle cx="80" cy="40" r="2" fill="black"/><!-- n=6, c=3: (20+6*10, 100 - 3*20) = (80, 40) --></svg></p><p>$c_n = (-1)^n \cdot \frac{n}{2}$</p><p><strong>Рисунок 35</strong></p><p><svg viewBox="0 0 120 160" xmlns="http://www.w3.org/2000/svg" width="120" height="160"><!-- Grid lines --><!-- Horizontal grid lines --><line x1="20" y1="10" x2="110" y2="10" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="20" x2="110" y2="20" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="30" x2="110" y2="30" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="40" x2="110" y2="40" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="50" x2="110" y2="50" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="60" x2="110" y2="60" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="70" x2="110" y2="70" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="80" x2="110" y2="80" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="90" x2="110" y2="90" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="100" x2="110" y2="100" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="110" x2="110" y2="110" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="120" x2="110" y2="120" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="130" x2="110" y2="130" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="140" x2="110" y2="140" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="150" x2="110" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Vertical grid lines --><line x1="20" y1="10" x2="20" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="30" y1="10" x2="30" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="40" y1="10" x2="40" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="50" y1="10" x2="50" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="60" y1="10" x2="60" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="70" y1="10" x2="70" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="80" y1="10" x2="80" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="90" y1="10" x2="90" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="100" y1="10" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="110" y1="10" x2="110" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Axes (Origin at 20, 100) --><line x1="20" y1="100" x2="110" y2="100" stroke="black" stroke-width="1.5"/><!-- X-axis --><line x1="20" y1="20" x2="20" y2="140" stroke="black" stroke-width="1.5"/><!-- Y-axis --><!-- Axis labels --><text x="15" y="105" font-family="sans-serif" font-size="8px">O</text><text x="110" y="105" font-family="sans-serif" font-size="8px">n</text><text x="10" y="25" font-family="sans-serif" font-size="8px">d<tspan baseline-shift="sub" font-size="6px">n</tspan></text><text x="10" y="95" font-family="sans-serif" font-size="8px">1-</text><!-- Y-label for 1. At y=100 - 1*10 = 90px. Text baseline is 90+5=95. --><text x="10" y="75" font-family="sans-serif" font-size="8px">3-</text><!-- Y-label for 3. At y=100 - 3*10 = 70px. Text baseline is 70+5=75. --><text x="27" y="110" font-family="sans-serif" font-size="8px">1</text><text x="57" y="110" font-family="sans-serif" font-size="8px">4</text><text x="77" y="110" font-family="sans-serif" font-size="8px">6</text><!-- Formula --><text x="50" y="40" font-family="sans-serif" font-size="8px">$d_n = -\frac{1}{2}(n-5)^2 + 4$</text><!-- Plotted points (OX=20, OY=100, unit_x_px=10, unit_y_px=10) --><circle cx="40" cy="105" r="2" fill="black"/><!-- n=2, d=-0.5: (20+2*10, 100 - (-0.5)*10) = (40, 105) --><circle cx="50" cy="80" r="2" fill="black"/><!-- n=3, d=2: (20+3*10, 100 - 2*10) = (50, 80) --><circle cx="60" cy="65" r="2" fill="black"/><!-- n=4, d=3.5: (20+4*10, 100 - 3.5*10) = (60, 65) --><circle cx="70" cy="60" r="2" fill="black"/><!-- n=5, d=4: (20+5*10, 100 - 4*10) = (70, 60) --><circle cx="80" cy="65" r="2" fill="black"/><!-- n=6, d=3.5: (20+6*10, 100 - 3.5*10) = (80, 65) --><circle cx="90" cy="80" r="2" fill="black"/><!-- n=7, d=2: (20+7*10, 100 - 2*10) = (90, 80) --><circle cx="100" cy="105" r="2" fill="black"/><!-- n=8, d=-0.5: (20+8*10, 100 - (-0.5)*10) = (100, 105) --></svg></p><p>$d_n = -\frac{1}{2}(n-5)^2 + 4$</p>" "img" => array:2 [ 0 => array:5 [ "name" => "317-1.jpg" "alt" => null "width" => "1573" "height" => 754 "path" => "/media/algebra_09/soltan-u19/0-1/317-1.webp?ts=1752834354" ] 1 => array:5 [ "name" => "317-2.jpg" "alt" => null "width" => "1356" "height" => 553 "path" => "/media/algebra_09/soltan-u19/0-1/317-2.webp?ts=1752834354" ] ] ] #original: array:7 [ "id" => 1276672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187} "task" => array:2 [ "refs" => "1108035" "type" => "task" ] "text" => "<p><strong>317.</strong> Установите, на каком из рисунков 32–35 изображен график:</p><p>а) возрастающей последовательности; б) убывающей последовательности.</p><p><strong>Рисунок 32</strong></p><p><svg viewBox="0 0 120 160" xmlns="http://www.w3.org/2000/svg" width="120" height="160"><!-- Grid lines --><!-- Horizontal grid lines --><line x1="20" y1="10" x2="100" y2="10" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="20" x2="100" y2="20" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="30" x2="100" y2="30" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="40" x2="100" y2="40" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="50" x2="100" y2="50" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="60" x2="100" y2="60" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="70" x2="100" y2="70" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="80" x2="100" y2="80" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="90" x2="100" y2="90" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="100" x2="100" y2="100" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="110" x2="100" y2="110" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="120" x2="100" y2="120" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="130" x2="100" y2="130" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="140" x2="100" y2="140" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="150" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Vertical grid lines --><line x1="20" y1="10" x2="20" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="30" y1="10" x2="30" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="40" y1="10" x2="40" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="50" y1="10" x2="50" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="60" y1="10" x2="60" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="70" y1="10" x2="70" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="80" y1="10" x2="80" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="90" y1="10" x2="90" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="100" y1="10" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Axes (Origin at 20, 100) --><line x1="20" y1="100" x2="100" y2="100" stroke="black" stroke-width="1.5"/><!-- X-axis --><line x1="20" y1="20" x2="20" y2="140" stroke="black" stroke-width="1.5"/><!-- Y-axis --><!-- Axis labels --><text x="15" y="105" font-family="sans-serif" font-size="8px">O</text><text x="100" y="105" font-family="sans-serif" font-size="8px">n</text><text x="10" y="25" font-family="sans-serif" font-size="8px">a<tspan baseline-shift="sub" font-size="6px">n</tspan></text><text x="10" y="85" font-family="sans-serif" font-size="8px">1-</text><!-- Y-label for 1. At y=100 - 1*20 = 80px. Text baseline is 80+5=85. --><text x="57" y="110" font-family="sans-serif" font-size="8px">4</text><text x="77" y="110" font-family="sans-serif" font-size="8px">6</text><!-- Formula (placed visually like in image) --><text x="50" y="40" font-family="sans-serif" font-size="8px">$a_n = \frac{n-3}{2}$</text><!-- Plotted points (OX=20, OY=100, unit_x_px=10, unit_y_px=20) --><circle cx="30" cy="120" r="2" fill="black"/><!-- n=1, a=-1: (20+1*10, 100 - (-1)*20) = (30, 120) --><circle cx="40" cy="110" r="2" fill="black"/><!-- n=2, a=-0.5: (20+2*10, 100 - (-0.5)*20) = (40, 110) --><circle cx="50" cy="100" r="2" fill="black"/><!-- n=3, a=0: (20+3*10, 100 - 0*20) = (50, 100) --><circle cx="60" cy="90" r="2" fill="black"/><!-- n=4, a=0.5: (20+4*10, 100 - 0.5*20) = (60, 90) --><circle cx="70" cy="80" r="2" fill="black"/><!-- n=5, a=1: (20+5*10, 100 - 1*20) = (70, 80) --><circle cx="80" cy="70" r="2" fill="black"/><!-- n=6, a=1.5: (20+6*10, 100 - 1.5*20) = (80, 70) --></svg></p><p>$a_n = \frac{n-3}{2}$</p><p><strong>Рисунок 33</strong></p><p><svg viewBox="0 0 120 160" xmlns="http://www.w3.org/2000/svg" width="120" height="160"><!-- Grid lines --><!-- Horizontal grid lines --><line x1="20" y1="10" x2="100" y2="10" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="20" x2="100" y2="20" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="30" x2="100" y2="30" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="40" x2="100" y2="40" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="50" x2="100" y2="50" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="60" x2="100" y2="60" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="70" x2="100" y2="70" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="80" x2="100" y2="80" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="90" x2="100" y2="90" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="100" x2="100" y2="100" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="110" x2="100" y2="110" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="120" x2="100" y2="120" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="130" x2="100" y2="130" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="140" x2="100" y2="140" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="150" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Vertical grid lines --><line x1="20" y1="10" x2="20" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="30" y1="10" x2="30" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="40" y1="10" x2="40" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="50" y1="10" x2="50" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="60" y1="10" x2="60" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="70" y1="10" x2="70" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="80" y1="10" x2="80" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="90" y1="10" x2="90" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="100" y1="10" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Axes (Origin at 20, 100) --><line x1="20" y1="100" x2="100" y2="100" stroke="black" stroke-width="1.5"/><!-- X-axis --><line x1="20" y1="20" x2="20" y2="140" stroke="black" stroke-width="1.5"/><!-- Y-axis --><!-- Axis labels --><text x="15" y="105" font-family="sans-serif" font-size="8px">O</text><text x="100" y="105" font-family="sans-serif" font-size="8px">n</text><text x="10" y="25" font-family="sans-serif" font-size="8px">b<tspan baseline-shift="sub" font-size="6px">n</tspan></text><text x="10" y="85" font-family="sans-serif" font-size="8px">1-</text><!-- Y-label for 1 --><text x="77" y="110" font-family="sans-serif" font-size="8px">6</text><!-- Formula --><text x="50" y="40" font-family="sans-serif" font-size="8px">$b_n = \frac{4}{n} - 2$</text><!-- Plotted points (OX=20, OY=100, unit_x_px=10, unit_y_px=20) --><circle cx="30" cy="60" r="2" fill="black"/><!-- n=1, b=2: (20+1*10, 100 - 2*20) = (30, 60) --><circle cx="40" cy="100" r="2" fill="black"/><!-- n=2, b=0: (20+2*10, 100 - 0*20) = (40, 100) --><circle cx="50" cy="113.33" r="2" fill="black"/><!-- n=3, b=-2/3: (20+3*10, 100 - (-2/3)*20) = (50, 113.33) --><circle cx="60" cy="120" r="2" fill="black"/><!-- n=4, b=-1: (20+4*10, 100 - (-1)*20) = (60, 120) --><circle cx="70" cy="124" r="2" fill="black"/><!-- n=5, b=-1.2: (20+5*10, 100 - (-1.2)*20) = (70, 124) --><circle cx="80" cy="126.67" r="2" fill="black"/><!-- n=6, b=-4/3: (20+6*10, 100 - (-4/3)*20) = (80, 126.67) --></svg></p><p>$b_n = \frac{4}{n} - 2$</p><p><strong>Рисунок 34</strong></p><p><svg viewBox="0 0 120 160" xmlns="http://www.w3.org/2000/svg" width="120" height="160"><!-- Grid lines --><!-- Horizontal grid lines --><line x1="20" y1="10" x2="100" y2="10" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="20" x2="100" y2="20" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="30" x2="100" y2="30" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="40" x2="100" y2="40" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="50" x2="100" y2="50" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="60" x2="100" y2="60" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="70" x2="100" y2="70" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="80" x2="100" y2="80" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="90" x2="100" y2="90" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="100" x2="100" y2="100" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="110" x2="100" y2="110" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="120" x2="100" y2="120" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="130" x2="100" y2="130" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="140" x2="100" y2="140" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="150" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Vertical grid lines --><line x1="20" y1="10" x2="20" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="30" y1="10" x2="30" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="40" y1="10" x2="40" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="50" y1="10" x2="50" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="60" y1="10" x2="60" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="70" y1="10" x2="70" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="80" y1="10" x2="80" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="90" y1="10" x2="90" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="100" y1="10" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Axes (Origin at 20, 100) --><line x1="20" y1="100" x2="100" y2="100" stroke="black" stroke-width="1.5"/><!-- X-axis --><line x1="20" y1="20" x2="20" y2="150" stroke="black" stroke-width="1.5"/><!-- Y-axis --><!-- Axis labels --><text x="15" y="105" font-family="sans-serif" font-size="8px">O</text><text x="100" y="105" font-family="sans-serif" font-size="8px">n</text><text x="10" y="25" font-family="sans-serif" font-size="8px">c<tspan baseline-shift="sub" font-size="6px">n</tspan></text><text x="10" y="85" font-family="sans-serif" font-size="8px">1-</text><!-- Y-label for 1 --><text x="37" y="110" font-family="sans-serif" font-size="8px">2</text><text x="57" y="110" font-family="sans-serif" font-size="8px">4</text><text x="77" y="110" font-family="sans-serif" font-size="8px">6</text><!-- Formula --><text x="50" y="40" font-family="sans-serif" font-size="8px">$c_n = (-1)^n \cdot \frac{n}{2}$</text><!-- Plotted points (OX=20, OY=100, unit_x_px=10, unit_y_px=20) --><circle cx="30" cy="110" r="2" fill="black"/><!-- n=1, c=-0.5: (20+1*10, 100 - (-0.5)*20) = (30, 110) --><circle cx="40" cy="80" r="2" fill="black"/><!-- n=2, c=1: (20+2*10, 100 - 1*20) = (40, 80) --><circle cx="50" cy="130" r="2" fill="black"/><!-- n=3, c=-1.5: (20+3*10, 100 - (-1.5)*20) = (50, 130) --><circle cx="60" cy="60" r="2" fill="black"/><!-- n=4, c=2: (20+4*10, 100 - 2*20) = (60, 60) --><circle cx="70" cy="150" r="2" fill="black"/><!-- n=5, c=-2.5: (20+5*10, 100 - (-2.5)*20) = (70, 150) --><circle cx="80" cy="40" r="2" fill="black"/><!-- n=6, c=3: (20+6*10, 100 - 3*20) = (80, 40) --></svg></p><p>$c_n = (-1)^n \cdot \frac{n}{2}$</p><p><strong>Рисунок 35</strong></p><p><svg viewBox="0 0 120 160" xmlns="http://www.w3.org/2000/svg" width="120" height="160"><!-- Grid lines --><!-- Horizontal grid lines --><line x1="20" y1="10" x2="110" y2="10" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="20" x2="110" y2="20" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="30" x2="110" y2="30" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="40" x2="110" y2="40" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="50" x2="110" y2="50" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="60" x2="110" y2="60" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="70" x2="110" y2="70" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="80" x2="110" y2="80" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="90" x2="110" y2="90" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="100" x2="110" y2="100" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="110" x2="110" y2="110" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="120" x2="110" y2="120" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="130" x2="110" y2="130" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="140" x2="110" y2="140" stroke="#ccc" stroke-width="0.5"/><line x1="20" y1="150" x2="110" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Vertical grid lines --><line x1="20" y1="10" x2="20" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="30" y1="10" x2="30" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="40" y1="10" x2="40" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="50" y1="10" x2="50" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="60" y1="10" x2="60" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="70" y1="10" x2="70" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="80" y1="10" x2="80" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="90" y1="10" x2="90" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="100" y1="10" x2="100" y2="150" stroke="#ccc" stroke-width="0.5"/><line x1="110" y1="10" x2="110" y2="150" stroke="#ccc" stroke-width="0.5"/><!-- Axes (Origin at 20, 100) --><line x1="20" y1="100" x2="110" y2="100" stroke="black" stroke-width="1.5"/><!-- X-axis --><line x1="20" y1="20" x2="20" y2="140" stroke="black" stroke-width="1.5"/><!-- Y-axis --><!-- Axis labels --><text x="15" y="105" font-family="sans-serif" font-size="8px">O</text><text x="110" y="105" font-family="sans-serif" font-size="8px">n</text><text x="10" y="25" font-family="sans-serif" font-size="8px">d<tspan baseline-shift="sub" font-size="6px">n</tspan></text><text x="10" y="95" font-family="sans-serif" font-size="8px">1-</text><!-- Y-label for 1. At y=100 - 1*10 = 90px. Text baseline is 90+5=95. --><text x="10" y="75" font-family="sans-serif" font-size="8px">3-</text><!-- Y-label for 3. At y=100 - 3*10 = 70px. Text baseline is 70+5=75. --><text x="27" y="110" font-family="sans-serif" font-size="8px">1</text><text x="57" y="110" font-family="sans-serif" font-size="8px">4</text><text x="77" y="110" font-family="sans-serif" font-size="8px">6</text><!-- Formula --><text x="50" y="40" font-family="sans-serif" font-size="8px">$d_n = -\frac{1}{2}(n-5)^2 + 4$</text><!-- Plotted points (OX=20, OY=100, unit_x_px=10, unit_y_px=10) --><circle cx="40" cy="105" r="2" fill="black"/><!-- n=2, d=-0.5: (20+2*10, 100 - (-0.5)*10) = (40, 105) --><circle cx="50" cy="80" r="2" fill="black"/><!-- n=3, d=2: (20+3*10, 100 - 2*10) = (50, 80) --><circle cx="60" cy="65" r="2" fill="black"/><!-- n=4, d=3.5: (20+4*10, 100 - 3.5*10) = (60, 65) --><circle cx="70" cy="60" r="2" fill="black"/><!-- n=5, d=4: (20+5*10, 100 - 4*10) = (70, 60) --><circle cx="80" cy="65" r="2" fill="black"/><!-- n=6, d=3.5: (20+6*10, 100 - 3.5*10) = (80, 65) --><circle cx="90" cy="80" r="2" fill="black"/><!-- n=7, d=2: (20+7*10, 100 - 2*10) = (90, 80) --><circle cx="100" cy="105" r="2" fill="black"/><!-- n=8, d=-0.5: (20+8*10, 100 - (-0.5)*10) = (100, 105) --></svg></p><p>$d_n = -\frac{1}{2}(n-5)^2 + 4$</p>" "img" => array:2 [ 0 => array:5 [ "name" => "317-1.jpg" "alt" => null "width" => "1573" "height" => 754 "path" => "/media/algebra_09/soltan-u19/0-1/317-1.webp?ts=1752834354" ] 1 => array:5 [ "name" => "317-2.jpg" "alt" => null "width" => "1356" "height" => 553 "path" => "/media/algebra_09/soltan-u19/0-1/317-2.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 {#1185 #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" => 1277963 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Edition {#1186 #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 {#1189 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108035" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "317-1.jpg" "alt" => null "width" => "612" "height" => 422 "path" => "/media/algebra_09/soltan-u19/1-1/317-1.webp?ts=1752831055" ] ] ] #original: array:6 [ "id" => 1277963 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195} "task" => array:2 [ "refs" => "1108035" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "317-1.jpg" "alt" => null "width" => "612" "height" => 422 "path" => "/media/algebra_09/soltan-u19/1-1/317-1.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 {#1193 #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" => 1553918 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Edition {#1194 #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 {#1197 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108035" "type" => "task" ] "text" => "<p><strong>а) возрастающей последовательности;</strong><br>Возрастающей называется последовательность, у которой каждый последующий член больше предыдущего, то есть для всех натуральных $n$ выполняется неравенство $x_{n+1} > x_n$. Для определения такой последовательности можно проанализировать разность $x_{n+1} - x_n$. Если эта разность положительна для всех $n$, последовательность возрастает.<br>Проверим каждую последовательность:<br>1. <em>Рисунок 32</em>: Последовательность задана формулой $a_n = \frac{n-3}{2}$. Найдем разность $a_{n+1} - a_n$:<br>$a_{n+1} - a_n = \frac{(n+1)-3}{2} - \frac{n-3}{2} = \frac{n-2}{2} - \frac{n-3}{2} = \frac{n-2-(n-3)}{2} = \frac{1}{2}$.<br>Поскольку разность $a_{n+1} - a_n = \frac{1}{2} > 0$ для любого натурального $n$, эта последовательность является возрастающей.<br>2. <em>Рисунок 33</em>: Последовательность $b_n = \frac{4}{n} - 2$. Разность $b_{n+1} - b_n = \left(\frac{4}{n+1} - 2\right) - \left(\frac{4}{n} - 2\right) = \frac{4}{n+1} - \frac{4}{n} = \frac{4n - 4(n+1)}{n(n+1)} = \frac{-4}{n(n+1)}$. Так как $n > 0$, разность всегда отрицательна, следовательно, последовательность убывает.<br>3. <em>Рисунок 34</em>: Последовательность $c_n = (-1)^n \cdot \frac{n}{2}$. Ее члены: $c_1=-0.5, c_2=1, c_3=-1.5, \dots$. Так как $c_2 > c_1$, а $c_3 < c_2$, последовательность не является возрастающей.<br>4. <em>Рисунок 35</em>: Последовательность $d_n = -\frac{1}{2}(n-5)^2 + 4$. Ее члены: $d_4=3.5, d_5=4, d_6=3.5$. Так как $d_5 > d_4$, а $d_6 < d_5$, последовательность не является возрастающей на всей числовой оси.<br>Таким образом, график возрастающей последовательности изображен на рисунке 32.<br>Ответ: Рисунок 32.</p><p><strong>б) убывающей последовательности.</strong><br>Убывающей называется последовательность, у которой каждый последующий член меньше предыдущего, то есть для всех натуральных $n$ выполняется неравенство $x_{n+1} < x_n$. Для определения такой последовательности можно проанализировать разность $x_{n+1} - x_n$. Если эта разность отрицательна для всех $n$, последовательность убывает.<br>Используя анализ из пункта а), мы можем сделать следующие выводы:<br>1. <em>Рисунок 32</em>: Последовательность $a_n$ является возрастающей.<br>2. <em>Рисунок 33</em>: Последовательность $b_n = \frac{4}{n} - 2$. Как было показано, разность $b_{n+1} - b_n = \frac{-4}{n(n+1)}$. Для любого натурального $n$, знаменатель $n(n+1)$ положителен, значит вся дробь отрицательна. Следовательно, $b_{n+1} < b_n$ для всех $n$, и последовательность является убывающей.<br>3. <em>Рисунок 34</em>: Последовательность $c_n$ не является монотонной (ни возрастающей, ни убывающей).<br>4. <em>Рисунок 35</em>: Последовательность $d_n$ не является монотонной.<br>Таким образом, график убывающей последовательности изображен на рисунке 33.<br>Ответ: Рисунок 33.</p>" ] #original: array:6 [ "id" => 1553918 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203} "task" => array:2 [ "refs" => "1108035" "type" => "task" ] "text" => "<p><strong>а) возрастающей последовательности;</strong><br>Возрастающей называется последовательность, у которой каждый последующий член больше предыдущего, то есть для всех натуральных $n$ выполняется неравенство $x_{n+1} > x_n$. Для определения такой последовательности можно проанализировать разность $x_{n+1} - x_n$. Если эта разность положительна для всех $n$, последовательность возрастает.<br>Проверим каждую последовательность:<br>1. <em>Рисунок 32</em>: Последовательность задана формулой $a_n = \frac{n-3}{2}$. Найдем разность $a_{n+1} - a_n$:<br>$a_{n+1} - a_n = \frac{(n+1)-3}{2} - \frac{n-3}{2} = \frac{n-2}{2} - \frac{n-3}{2} = \frac{n-2-(n-3)}{2} = \frac{1}{2}$.<br>Поскольку разность $a_{n+1} - a_n = \frac{1}{2} > 0$ для любого натурального $n$, эта последовательность является возрастающей.<br>2. <em>Рисунок 33</em>: Последовательность $b_n = \frac{4}{n} - 2$. Разность $b_{n+1} - b_n = \left(\frac{4}{n+1} - 2\right) - \left(\frac{4}{n} - 2\right) = \frac{4}{n+1} - \frac{4}{n} = \frac{4n - 4(n+1)}{n(n+1)} = \frac{-4}{n(n+1)}$. Так как $n > 0$, разность всегда отрицательна, следовательно, последовательность убывает.<br>3. <em>Рисунок 34</em>: Последовательность $c_n = (-1)^n \cdot \frac{n}{2}$. Ее члены: $c_1=-0.5, c_2=1, c_3=-1.5, \dots$. Так как $c_2 > c_1$, а $c_3 < c_2$, последовательность не является возрастающей.<br>4. <em>Рисунок 35</em>: Последовательность $d_n = -\frac{1}{2}(n-5)^2 + 4$. Ее члены: $d_4=3.5, d_5=4, d_6=3.5$. Так как $d_5 > d_4$, а $d_6 < d_5$, последовательность не является возрастающей на всей числовой оси.<br>Таким образом, график возрастающей последовательности изображен на рисунке 32.<br>Ответ: Рисунок 32.</p><p><strong>б) убывающей последовательности.</strong><br>Убывающей называется последовательность, у которой каждый последующий член меньше предыдущего, то есть для всех натуральных $n$ выполняется неравенство $x_{n+1} < x_n$. Для определения такой последовательности можно проанализировать разность $x_{n+1} - x_n$. Если эта разность отрицательна для всех $n$, последовательность убывает.<br>Используя анализ из пункта а), мы можем сделать следующие выводы:<br>1. <em>Рисунок 32</em>: Последовательность $a_n$ является возрастающей.<br>2. <em>Рисунок 33</em>: Последовательность $b_n = \frac{4}{n} - 2$. Как было показано, разность $b_{n+1} - b_n = \frac{-4}{n(n+1)}$. Для любого натурального $n$, знаменатель $n(n+1)$ положителен, значит вся дробь отрицательна. Следовательно, $b_{n+1} < b_n$ для всех $n$, и последовательность является убывающей.<br>3. <em>Рисунок 34</em>: Последовательность $c_n$ не является монотонной (ни возрастающей, ни убывающей).<br>4. <em>Рисунок 35</em>: Последовательность $d_n$ не является монотонной.<br>Таким образом, график убывающей последовательности изображен на рисунке 33.<br>Ответ: Рисунок 33.</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" => "1108036" "type" => "task" ] "previous" => array:2 [ "refs" => "1108034" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Book {#1175 #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 {#1168 #items: array:1 [ 0 => App\Models\Term {#1167 #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 [ …6] "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 {#1174 #items: array:1 [ 0 => App\Models\Term {#1176 #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 {#1172 #items: array:1 [ 0 => App\Models\Term {#1170 #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 {#1173 #items: array:3 [ 0 => App\Models\Term {#1171 #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 {#1201 #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 {#1202 #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 {#1204 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ 0 => App\Models\Term {#1206 #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 {#1207 #items: array:1 [ 0 => App\Models\Term {#1208 #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 {#1209 #items: array:1 [ 0 => App\Models\Term {#1210 #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 {#1211 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1213 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1214 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1215 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216 #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 {#1217 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220 #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 {#1168} "field_class" => Illuminate\Database\Eloquent\Collection {#1174} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1172} "field_author" => Illuminate\Database\Eloquent\Collection {#1173} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1204} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1205} "field_country" => Illuminate\Database\Eloquent\Collection {#1207} "field_city" => Illuminate\Database\Eloquent\Collection {#1209} "field_series" => Illuminate\Database\Eloquent\Collection {#1211} "field_umk" => Illuminate\Database\Eloquent\Collection {#1212} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1213} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1214} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1215} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216} "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 {#1217} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:1 [ 0 => App\Models\BookPage {#1227 #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" => 1097838 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-96" "field_display_title" => "96" "field_folder" => "folder1" "field_image_name" => "96" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1229 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:1 [ 0 => App\Models\Book {#1175} ] #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 {#1231 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ 0 => App\Models\Element {#1240 #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" => 1260989 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1242 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260989 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1242 …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" => "1097839" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097837" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1426 #items: array:6 [ 0 => App\Models\Task {#1438 #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" => 1108030 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/vopr-96" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1439 …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 {#1440 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1441 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1442 …2} "content" => Illuminate\Database\Eloquent\Collection {#1452 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1459 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108030 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/vopr-96" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1439 …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 {#1440 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1441 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1442 …2} "content" => Illuminate\Database\Eloquent\Collection {#1452 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1459 …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 {#1443 #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" => 1108031 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-313" "field_display_title" => "313" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1445 …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 {#1444 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1450 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1449 …2} "content" => Illuminate\Database\Eloquent\Collection {#1447 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1454 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108031 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-313" "field_display_title" => "313" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1445 …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 {#1444 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1450 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1449 …2} "content" => Illuminate\Database\Eloquent\Collection {#1447 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1454 …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 {#1446 #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" => 1108032 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-314" "field_display_title" => "314" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1448 …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 {#1456 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1457 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1458 …2} "content" => Illuminate\Database\Eloquent\Collection {#1451 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1479 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108032 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-314" "field_display_title" => "314" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1448 …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 {#1456 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1457 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1458 …2} "content" => Illuminate\Database\Eloquent\Collection {#1451 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1479 …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 {#1460 #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" => 1108033 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" 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№317 (с. 96)
Условие. №317 (с. 96)
скриншот условия
317. Установите, на каком из рисунков 32–35 изображен график:
а) возрастающей последовательности; б) убывающей последовательности.
Рисунок 32
$a_n = \frac{n-3}{2}$
Рисунок 33
$b_n = \frac{4}{n} - 2$
Рисунок 34
$c_n = (-1)^n \cdot \frac{n}{2}$
Рисунок 35
$d_n = -\frac{1}{2}(n-5)^2 + 4$
Решение. №317 (с. 96)
Решение 2 (rus). №317 (с. 96)
а) возрастающей последовательности;
Возрастающей называется последовательность, у которой каждый последующий член больше предыдущего, то есть для всех натуральных $n$ выполняется неравенство $x_{n+1} > x_n$. Для определения такой последовательности можно проанализировать разность $x_{n+1} - x_n$. Если эта разность положительна для всех $n$, последовательность возрастает.
Проверим каждую последовательность:
1. Рисунок 32: Последовательность задана формулой $a_n = \frac{n-3}{2}$. Найдем разность $a_{n+1} - a_n$:
$a_{n+1} - a_n = \frac{(n+1)-3}{2} - \frac{n-3}{2} = \frac{n-2}{2} - \frac{n-3}{2} = \frac{n-2-(n-3)}{2} = \frac{1}{2}$.
Поскольку разность $a_{n+1} - a_n = \frac{1}{2} > 0$ для любого натурального $n$, эта последовательность является возрастающей.
2. Рисунок 33: Последовательность $b_n = \frac{4}{n} - 2$. Разность $b_{n+1} - b_n = \left(\frac{4}{n+1} - 2\right) - \left(\frac{4}{n} - 2\right) = \frac{4}{n+1} - \frac{4}{n} = \frac{4n - 4(n+1)}{n(n+1)} = \frac{-4}{n(n+1)}$. Так как $n > 0$, разность всегда отрицательна, следовательно, последовательность убывает.
3. Рисунок 34: Последовательность $c_n = (-1)^n \cdot \frac{n}{2}$. Ее члены: $c_1=-0.5, c_2=1, c_3=-1.5, \dots$. Так как $c_2 > c_1$, а $c_3 < c_2$, последовательность не является возрастающей.
4. Рисунок 35: Последовательность $d_n = -\frac{1}{2}(n-5)^2 + 4$. Ее члены: $d_4=3.5, d_5=4, d_6=3.5$. Так как $d_5 > d_4$, а $d_6 < d_5$, последовательность не является возрастающей на всей числовой оси.
Таким образом, график возрастающей последовательности изображен на рисунке 32.
Ответ: Рисунок 32.
б) убывающей последовательности.
Убывающей называется последовательность, у которой каждый последующий член меньше предыдущего, то есть для всех натуральных $n$ выполняется неравенство $x_{n+1} < x_n$. Для определения такой последовательности можно проанализировать разность $x_{n+1} - x_n$. Если эта разность отрицательна для всех $n$, последовательность убывает.
Используя анализ из пункта а), мы можем сделать следующие выводы:
1. Рисунок 32: Последовательность $a_n$ является возрастающей.
2. Рисунок 33: Последовательность $b_n = \frac{4}{n} - 2$. Как было показано, разность $b_{n+1} - b_n = \frac{-4}{n(n+1)}$. Для любого натурального $n$, знаменатель $n(n+1)$ положителен, значит вся дробь отрицательна. Следовательно, $b_{n+1} < b_n$ для всех $n$, и последовательность является убывающей.
3. Рисунок 34: Последовательность $c_n$ не является монотонной (ни возрастающей, ни убывающей).
4. Рисунок 35: Последовательность $d_n$ не является монотонной.
Таким образом, график убывающей последовательности изображен на рисунке 33.
Ответ: Рисунок 33.
Другие задания:
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