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Номер 361, страница 107 - 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. 14. Метод математической индукции - номер 361, страница 107.
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" => 1108083 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "107" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-361" "field_display_title" => "361" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045} "field_author" => Illuminate\Database\Eloquent\Collection {#1047} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052} "field_country" => Illuminate\Database\Eloquent\Collection {#1054} "field_city" => Illuminate\Database\Eloquent\Collection {#1056} "field_series" => Illuminate\Database\Eloquent\Collection {#1058} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1138 #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 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 #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 {#1074} "field_class" => Illuminate\Database\Eloquent\Collection {#1076} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078} "field_author" => Illuminate\Database\Eloquent\Collection {#1080} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085} "field_country" => Illuminate\Database\Eloquent\Collection {#1087} "field_city" => Illuminate\Database\Eloquent\Collection {#1089} "field_series" => Illuminate\Database\Eloquent\Collection {#1091} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#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" => 1107655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. Метод математической индукции" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "101" "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 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 #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 {#1106} "field_class" => Illuminate\Database\Eloquent\Collection {#1108} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110} "field_author" => Illuminate\Database\Eloquent\Collection {#1112} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117} "field_country" => Illuminate\Database\Eloquent\Collection {#1119} "field_city" => Illuminate\Database\Eloquent\Collection {#1121} "field_series" => Illuminate\Database\Eloquent\Collection {#1123} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#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 #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 {#1136 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 #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 {#1135} "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} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137} ] #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" => 1107655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "14. Метод математической индукции" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "101" "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 {#1148 #items: array:3 [ 0 => App\Models\Element {#1158 #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" => 1276716 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Edition {#1159 #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 {#1161 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1161} "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 {#1162} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164} "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" => "1108083" "type" => "task" ] "text" => "<p><strong>361.</strong> Исследуйте, при каких значениях $n \in \mathbb{N}$ верно неравенство:</p><p><strong>a)</strong> $2^n > n^2 + 4n + 5$;</p><p><strong>б)</strong> $3^n > 2^n + 7n$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "361-1.jpg" "alt" => null "width" => "1451" "height" => 209 "path" => "/media/algebra_09/soltan-u19/0-1/361-1.webp?ts=1752834398" ] ] ] #original: array:7 [ "id" => 1276716 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167} "task" => array:2 [ "refs" => "1108083" "type" => "task" ] "text" => "<p><strong>361.</strong> Исследуйте, при каких значениях $n \in \mathbb{N}$ верно неравенство:</p><p><strong>a)</strong> $2^n > n^2 + 4n + 5$;</p><p><strong>б)</strong> $3^n > 2^n + 7n$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "361-1.jpg" "alt" => null "width" => "1451" "height" => 209 "path" => "/media/algebra_09/soltan-u19/0-1/361-1.webp?ts=1752834398" ] ] ] #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 {#1165 #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" => 1278007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1166 #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 {#1169 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1170 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169} "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 {#1170} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172} "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" => "1108083" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "361-1.jpg" "alt" => null "width" => "1350" "height" => 1048 "path" => "/media/algebra_09/soltan-u19/1-1/361-1.webp?ts=1752831112" ] 1 => array:5 [ "name" => "361-2.jpg" "alt" => null "width" => "1396" "height" => 1080 "path" => "/media/algebra_09/soltan-u19/1-1/361-2.webp?ts=1752831112" ] ] ] #original: array:6 [ "id" => 1278007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1108083" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "361-1.jpg" "alt" => null "width" => "1350" "height" => 1048 "path" => "/media/algebra_09/soltan-u19/1-1/361-1.webp?ts=1752831112" ] 1 => array:5 [ "name" => "361-2.jpg" "alt" => null "width" => "1396" "height" => 1080 "path" => "/media/algebra_09/soltan-u19/1-1/361-2.webp?ts=1752831112" ] ] ] #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 {#1173 #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" => 1553962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Edition {#1174 #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 {#1177 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1177} "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 {#1178} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180} "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" => "1108083" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Исследуем, при каких натуральных значениях $n$ верно неравенство $2^n > n^2 + 4n + 5$.</p><p>Для решения задачи применим метод перебора для малых значений $n$, а затем докажем общую закономерность с помощью математической индукции.</p><p>Проверим неравенство для нескольких первых натуральных значений $n$:</p><p>При $n=1$: $2^1 = 2$; $1^2 + 4(1) + 5 = 10$. Неравенство $2 > 10$ неверно.</p><p>При $n=2$: $2^2 = 4$; $2^2 + 4(2) + 5 = 17$. Неравенство $4 > 17$ неверно.</p><p>При $n=3$: $2^3 = 8$; $3^2 + 4(3) + 5 = 26$. Неравенство $8 > 26$ неверно.</p><p>При $n=4$: $2^4 = 16$; $4^2 + 4(4) + 5 = 37$. Неравенство $16 > 37$ неверно.</p><p>При $n=5$: $2^5 = 32$; $5^2 + 4(5) + 5 = 50$. Неравенство $32 > 50$ неверно.</p><p>При $n=6$: $2^6 = 64$; $6^2 + 4(6) + 5 = 65$. Неравенство $64 > 65$ неверно.</p><p>При $n=7$: $2^7 = 128$; $7^2 + 4(7) + 5 = 49 + 28 + 5 = 82$. Неравенство $128 > 82$ верно.</p><p>При $n=8$: $2^8 = 256$; $8^2 + 4(8) + 5 = 64 + 32 + 5 = 101$. Неравенство $256 > 101$ верно.</p><p>Мы видим, что, начиная с $n=7$, левая часть неравенства (экспоненциальная функция) растет значительно быстрее правой (квадратичная функция). Предположим, что неравенство верно для всех натуральных $n \ge 7$, и докажем это методом математической индукции.</p><p><strong>База индукции:</strong> Для $n=7$ неравенство $128 > 82$ верно, что мы уже установили.</p><p><strong>Шаг индукции:</strong> Предположим, что неравенство верно для некоторого натурального числа $k \ge 7$, то есть выполнено $2^k > k^2 + 4k + 5$. Это наше предположение индукции.</p><p>Докажем, что из этого следует верность неравенства для $n = k+1$, то есть $2^{k+1} > (k+1)^2 + 4(k+1) + 5$.</p><p>Рассмотрим левую часть неравенства для $k+1$ и используем предположение индукции:</p><p>$2^{k+1} = 2 \cdot 2^k > 2(k^2 + 4k + 5) = 2k^2 + 8k + 10$.</p><p>Теперь рассмотрим правую часть неравенства для $k+1$:</p><p>$(k+1)^2 + 4(k+1) + 5 = (k^2 + 2k + 1) + (4k + 4) + 5 = k^2 + 6k + 10$.</p><p>Нам нужно доказать, что $2^{k+1}$ больше, чем $k^2 + 6k + 10$. Мы уже показали, что $2^{k+1} > 2k^2 + 8k + 10$. Если мы докажем, что $2k^2 + 8k + 10 > k^2 + 6k + 10$, то шаг индукции будет доказан.</p><p>Сравним $2k^2 + 8k + 10$ и $k^2 + 6k + 10$:</p><p>$2k^2 + 8k + 10 - (k^2 + 6k + 10) = k^2 + 2k$.</p><p>Так как мы рассматриваем $k \ge 7$, то $k$ является положительным числом, и, следовательно, $k^2 + 2k > 0$.</p><p>Это означает, что $2k^2 + 8k + 10 > k^2 + 6k + 10$.</p><p>Таким образом, мы построили следующую цепочку неравенств: $2^{k+1} > 2k^2 + 8k + 10 > k^2 + 6k + 10 = (k+1)^2 + 4(k+1) + 5$.</p><p>Шаг индукции доказан. По принципу математической индукции, неравенство $2^n > n^2 + 4n + 5$ верно для всех натуральных чисел $n \ge 7$.</p><p><strong>Ответ:</strong> для всех натуральных $n \ge 7$.</p><p><strong>б)</strong></p><p>Исследуем, при каких натуральных значениях $n$ верно неравенство $3^n > 2^n + 7n$.</p><p>Как и в предыдущем пункте, проверим неравенство для малых значений $n$ и затем воспользуемся методом математической индукции.</p><p>Проверка для первых нескольких натуральных значений $n$:</p><p>При $n=1$: $3^1 = 3$; $2^1 + 7(1) = 9$. Неравенство $3 > 9$ неверно.</p><p>При $n=2$: $3^2 = 9$; $2^2 + 7(2) = 18$. Неравенство $9 > 18$ неверно.</p><p>При $n=3$: $3^3 = 27$; $2^3 + 7(3) = 29$. Неравенство $27 > 29$ неверно.</p><p>При $n=4$: $3^4 = 81$; $2^4 + 7(4) = 16 + 28 = 44$. Неравенство $81 > 44$ верно.</p><p>При $n=5$: $3^5 = 243$; $2^5 + 7(5) = 32 + 35 = 67$. Неравенство $243 > 67$ верно.</p><p>Можно предположить, что неравенство верно для всех натуральных $n \ge 4$. Докажем это утверждение.</p><p><strong>База индукции:</strong> Для $n=4$ неравенство $81 > 44$ верно.</p><p><strong>Шаг индукции:</strong> Предположим, что неравенство верно для некоторого натурального числа $k \ge 4$, то есть $3^k > 2^k + 7k$.</p><p>Докажем, что неравенство верно и для $n = k+1$: $3^{k+1} > 2^{k+1} + 7(k+1)$.</p><p>Используя предположение индукции, преобразуем левую часть:</p><p>$3^{k+1} = 3 \cdot 3^k > 3(2^k + 7k) = 3 \cdot 2^k + 21k$.</p><p>Правая часть неравенства для $k+1$ равна:</p><p>$2^{k+1} + 7(k+1) = 2 \cdot 2^k + 7k + 7$.</p><p>Теперь докажем, что $3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7$.</p><p>Для этого рассмотрим их разность:</p><p>$(3 \cdot 2^k + 21k) - (2 \cdot 2^k + 7k + 7) = (3 \cdot 2^k - 2 \cdot 2^k) + (21k - 7k) - 7 = 2^k + 14k - 7$.</p><p>Нам нужно показать, что $2^k + 14k - 7 > 0$ для всех $k \ge 4$.</p><p>При $k=4$: $2^4 + 14(4) - 7 = 16 + 56 - 7 = 65 > 0$.</p><p>Функция $f(k) = 2^k + 14k - 7$ является возрастающей для $k \ge 1$, так как оба слагаемых $2^k$ и $14k$ являются возрастающими функциями. Следовательно, если неравенство верно для $k=4$, оно будет верно и для всех $k > 4$.</p><p>Таким образом, мы доказали, что $3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7$.</p><p>Получаем цепочку неравенств: $3^{k+1} > 3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7 = 2^{k+1} + 7(k+1)$.</p><p>Шаг индукции доказан. По принципу математической индукции, неравенство $3^n > 2^n + 7n$ верно для всех натуральных чисел $n \ge 4$.</p><p><strong>Ответ:</strong> для всех натуральных $n \ge 4$.</p>" ] #original: array:6 [ "id" => 1553962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183} "task" => array:2 [ "refs" => "1108083" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Исследуем, при каких натуральных значениях $n$ верно неравенство $2^n > n^2 + 4n + 5$.</p><p>Для решения задачи применим метод перебора для малых значений $n$, а затем докажем общую закономерность с помощью математической индукции.</p><p>Проверим неравенство для нескольких первых натуральных значений $n$:</p><p>При $n=1$: $2^1 = 2$; $1^2 + 4(1) + 5 = 10$. Неравенство $2 > 10$ неверно.</p><p>При $n=2$: $2^2 = 4$; $2^2 + 4(2) + 5 = 17$. Неравенство $4 > 17$ неверно.</p><p>При $n=3$: $2^3 = 8$; $3^2 + 4(3) + 5 = 26$. Неравенство $8 > 26$ неверно.</p><p>При $n=4$: $2^4 = 16$; $4^2 + 4(4) + 5 = 37$. Неравенство $16 > 37$ неверно.</p><p>При $n=5$: $2^5 = 32$; $5^2 + 4(5) + 5 = 50$. Неравенство $32 > 50$ неверно.</p><p>При $n=6$: $2^6 = 64$; $6^2 + 4(6) + 5 = 65$. Неравенство $64 > 65$ неверно.</p><p>При $n=7$: $2^7 = 128$; $7^2 + 4(7) + 5 = 49 + 28 + 5 = 82$. Неравенство $128 > 82$ верно.</p><p>При $n=8$: $2^8 = 256$; $8^2 + 4(8) + 5 = 64 + 32 + 5 = 101$. Неравенство $256 > 101$ верно.</p><p>Мы видим, что, начиная с $n=7$, левая часть неравенства (экспоненциальная функция) растет значительно быстрее правой (квадратичная функция). Предположим, что неравенство верно для всех натуральных $n \ge 7$, и докажем это методом математической индукции.</p><p><strong>База индукции:</strong> Для $n=7$ неравенство $128 > 82$ верно, что мы уже установили.</p><p><strong>Шаг индукции:</strong> Предположим, что неравенство верно для некоторого натурального числа $k \ge 7$, то есть выполнено $2^k > k^2 + 4k + 5$. Это наше предположение индукции.</p><p>Докажем, что из этого следует верность неравенства для $n = k+1$, то есть $2^{k+1} > (k+1)^2 + 4(k+1) + 5$.</p><p>Рассмотрим левую часть неравенства для $k+1$ и используем предположение индукции:</p><p>$2^{k+1} = 2 \cdot 2^k > 2(k^2 + 4k + 5) = 2k^2 + 8k + 10$.</p><p>Теперь рассмотрим правую часть неравенства для $k+1$:</p><p>$(k+1)^2 + 4(k+1) + 5 = (k^2 + 2k + 1) + (4k + 4) + 5 = k^2 + 6k + 10$.</p><p>Нам нужно доказать, что $2^{k+1}$ больше, чем $k^2 + 6k + 10$. Мы уже показали, что $2^{k+1} > 2k^2 + 8k + 10$. Если мы докажем, что $2k^2 + 8k + 10 > k^2 + 6k + 10$, то шаг индукции будет доказан.</p><p>Сравним $2k^2 + 8k + 10$ и $k^2 + 6k + 10$:</p><p>$2k^2 + 8k + 10 - (k^2 + 6k + 10) = k^2 + 2k$.</p><p>Так как мы рассматриваем $k \ge 7$, то $k$ является положительным числом, и, следовательно, $k^2 + 2k > 0$.</p><p>Это означает, что $2k^2 + 8k + 10 > k^2 + 6k + 10$.</p><p>Таким образом, мы построили следующую цепочку неравенств: $2^{k+1} > 2k^2 + 8k + 10 > k^2 + 6k + 10 = (k+1)^2 + 4(k+1) + 5$.</p><p>Шаг индукции доказан. По принципу математической индукции, неравенство $2^n > n^2 + 4n + 5$ верно для всех натуральных чисел $n \ge 7$.</p><p><strong>Ответ:</strong> для всех натуральных $n \ge 7$.</p><p><strong>б)</strong></p><p>Исследуем, при каких натуральных значениях $n$ верно неравенство $3^n > 2^n + 7n$.</p><p>Как и в предыдущем пункте, проверим неравенство для малых значений $n$ и затем воспользуемся методом математической индукции.</p><p>Проверка для первых нескольких натуральных значений $n$:</p><p>При $n=1$: $3^1 = 3$; $2^1 + 7(1) = 9$. Неравенство $3 > 9$ неверно.</p><p>При $n=2$: $3^2 = 9$; $2^2 + 7(2) = 18$. Неравенство $9 > 18$ неверно.</p><p>При $n=3$: $3^3 = 27$; $2^3 + 7(3) = 29$. Неравенство $27 > 29$ неверно.</p><p>При $n=4$: $3^4 = 81$; $2^4 + 7(4) = 16 + 28 = 44$. Неравенство $81 > 44$ верно.</p><p>При $n=5$: $3^5 = 243$; $2^5 + 7(5) = 32 + 35 = 67$. Неравенство $243 > 67$ верно.</p><p>Можно предположить, что неравенство верно для всех натуральных $n \ge 4$. Докажем это утверждение.</p><p><strong>База индукции:</strong> Для $n=4$ неравенство $81 > 44$ верно.</p><p><strong>Шаг индукции:</strong> Предположим, что неравенство верно для некоторого натурального числа $k \ge 4$, то есть $3^k > 2^k + 7k$.</p><p>Докажем, что неравенство верно и для $n = k+1$: $3^{k+1} > 2^{k+1} + 7(k+1)$.</p><p>Используя предположение индукции, преобразуем левую часть:</p><p>$3^{k+1} = 3 \cdot 3^k > 3(2^k + 7k) = 3 \cdot 2^k + 21k$.</p><p>Правая часть неравенства для $k+1$ равна:</p><p>$2^{k+1} + 7(k+1) = 2 \cdot 2^k + 7k + 7$.</p><p>Теперь докажем, что $3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7$.</p><p>Для этого рассмотрим их разность:</p><p>$(3 \cdot 2^k + 21k) - (2 \cdot 2^k + 7k + 7) = (3 \cdot 2^k - 2 \cdot 2^k) + (21k - 7k) - 7 = 2^k + 14k - 7$.</p><p>Нам нужно показать, что $2^k + 14k - 7 > 0$ для всех $k \ge 4$.</p><p>При $k=4$: $2^4 + 14(4) - 7 = 16 + 56 - 7 = 65 > 0$.</p><p>Функция $f(k) = 2^k + 14k - 7$ является возрастающей для $k \ge 1$, так как оба слагаемых $2^k$ и $14k$ являются возрастающими функциями. Следовательно, если неравенство верно для $k=4$, оно будет верно и для всех $k > 4$.</p><p>Таким образом, мы доказали, что $3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7$.</p><p>Получаем цепочку неравенств: $3^{k+1} > 3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7 = 2^{k+1} + 7(k+1)$.</p><p>Шаг индукции доказан. По принципу математической индукции, неравенство $3^n > 2^n + 7n$ верно для всех натуральных чисел $n \ge 4$.</p><p><strong>Ответ:</strong> для всех натуральных $n \ge 4$.</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" => "1108084" "type" => "task" ] "previous" => array:2 [ "refs" => "1108082" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1155 #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 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"algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\Term {#1146 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:1 [ 0 => App\Models\Term {#1144 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:3 [ 0 => App\Models\Term {#1143 #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 {#1149 #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 {#1147 #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 {#1154 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Term {#1152 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" …2 ] "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 {#1150 #items: array:1 [ 0 => App\Models\Term {#1153 #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 {#1151 #items: array:1 [ 0 => App\Models\Term {#1181 #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 {#1182 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188 #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 {#1189 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192 #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 {#1141} "field_class" => Illuminate\Database\Eloquent\Collection {#1139} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145} "field_author" => Illuminate\Database\Eloquent\Collection {#1142} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1154} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1156} "field_country" => Illuminate\Database\Eloquent\Collection {#1150} "field_city" => Illuminate\Database\Eloquent\Collection {#1151} "field_series" => Illuminate\Database\Eloquent\Collection {#1182} "field_umk" => Illuminate\Database\Eloquent\Collection {#1184} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188} "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 {#1189} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192} "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 {#1200 #items: array:1 [ 0 => App\Models\BookPage {#1199 #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" => 1097849 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "107" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-107" "field_display_title" => "107" "field_folder" => "folder1" "field_image_name" => "107" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1201 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Book {#1155} ] #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 {#1203 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Element {#1212 #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" => 1261000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …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" => "1097850" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097848" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1369 #items: array:3 [ 0 => App\Models\Task {#1379 #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" => 1108082 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "106" "field_page_end" => "107" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-360" "field_display_title" => "360" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1380 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => 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№361 (с. 107)
Условие. №361 (с. 107)
скриншот условия
361. Исследуйте, при каких значениях $n \in \mathbb{N}$ верно неравенство:
a) $2^n > n^2 + 4n + 5$;
б) $3^n > 2^n + 7n$.
Решение. №361 (с. 107)
Решение 2 (rus). №361 (с. 107)
а)
Исследуем, при каких натуральных значениях $n$ верно неравенство $2^n > n^2 + 4n + 5$.
Для решения задачи применим метод перебора для малых значений $n$, а затем докажем общую закономерность с помощью математической индукции.
Проверим неравенство для нескольких первых натуральных значений $n$:
При $n=1$: $2^1 = 2$; $1^2 + 4(1) + 5 = 10$. Неравенство $2 > 10$ неверно.
При $n=2$: $2^2 = 4$; $2^2 + 4(2) + 5 = 17$. Неравенство $4 > 17$ неверно.
При $n=3$: $2^3 = 8$; $3^2 + 4(3) + 5 = 26$. Неравенство $8 > 26$ неверно.
При $n=4$: $2^4 = 16$; $4^2 + 4(4) + 5 = 37$. Неравенство $16 > 37$ неверно.
При $n=5$: $2^5 = 32$; $5^2 + 4(5) + 5 = 50$. Неравенство $32 > 50$ неверно.
При $n=6$: $2^6 = 64$; $6^2 + 4(6) + 5 = 65$. Неравенство $64 > 65$ неверно.
При $n=7$: $2^7 = 128$; $7^2 + 4(7) + 5 = 49 + 28 + 5 = 82$. Неравенство $128 > 82$ верно.
При $n=8$: $2^8 = 256$; $8^2 + 4(8) + 5 = 64 + 32 + 5 = 101$. Неравенство $256 > 101$ верно.
Мы видим, что, начиная с $n=7$, левая часть неравенства (экспоненциальная функция) растет значительно быстрее правой (квадратичная функция). Предположим, что неравенство верно для всех натуральных $n \ge 7$, и докажем это методом математической индукции.
База индукции: Для $n=7$ неравенство $128 > 82$ верно, что мы уже установили.
Шаг индукции: Предположим, что неравенство верно для некоторого натурального числа $k \ge 7$, то есть выполнено $2^k > k^2 + 4k + 5$. Это наше предположение индукции.
Докажем, что из этого следует верность неравенства для $n = k+1$, то есть $2^{k+1} > (k+1)^2 + 4(k+1) + 5$.
Рассмотрим левую часть неравенства для $k+1$ и используем предположение индукции:
$2^{k+1} = 2 \cdot 2^k > 2(k^2 + 4k + 5) = 2k^2 + 8k + 10$.
Теперь рассмотрим правую часть неравенства для $k+1$:
$(k+1)^2 + 4(k+1) + 5 = (k^2 + 2k + 1) + (4k + 4) + 5 = k^2 + 6k + 10$.
Нам нужно доказать, что $2^{k+1}$ больше, чем $k^2 + 6k + 10$. Мы уже показали, что $2^{k+1} > 2k^2 + 8k + 10$. Если мы докажем, что $2k^2 + 8k + 10 > k^2 + 6k + 10$, то шаг индукции будет доказан.
Сравним $2k^2 + 8k + 10$ и $k^2 + 6k + 10$:
$2k^2 + 8k + 10 - (k^2 + 6k + 10) = k^2 + 2k$.
Так как мы рассматриваем $k \ge 7$, то $k$ является положительным числом, и, следовательно, $k^2 + 2k > 0$.
Это означает, что $2k^2 + 8k + 10 > k^2 + 6k + 10$.
Таким образом, мы построили следующую цепочку неравенств: $2^{k+1} > 2k^2 + 8k + 10 > k^2 + 6k + 10 = (k+1)^2 + 4(k+1) + 5$.
Шаг индукции доказан. По принципу математической индукции, неравенство $2^n > n^2 + 4n + 5$ верно для всех натуральных чисел $n \ge 7$.
Ответ: для всех натуральных $n \ge 7$.
б)
Исследуем, при каких натуральных значениях $n$ верно неравенство $3^n > 2^n + 7n$.
Как и в предыдущем пункте, проверим неравенство для малых значений $n$ и затем воспользуемся методом математической индукции.
Проверка для первых нескольких натуральных значений $n$:
При $n=1$: $3^1 = 3$; $2^1 + 7(1) = 9$. Неравенство $3 > 9$ неверно.
При $n=2$: $3^2 = 9$; $2^2 + 7(2) = 18$. Неравенство $9 > 18$ неверно.
При $n=3$: $3^3 = 27$; $2^3 + 7(3) = 29$. Неравенство $27 > 29$ неверно.
При $n=4$: $3^4 = 81$; $2^4 + 7(4) = 16 + 28 = 44$. Неравенство $81 > 44$ верно.
При $n=5$: $3^5 = 243$; $2^5 + 7(5) = 32 + 35 = 67$. Неравенство $243 > 67$ верно.
Можно предположить, что неравенство верно для всех натуральных $n \ge 4$. Докажем это утверждение.
База индукции: Для $n=4$ неравенство $81 > 44$ верно.
Шаг индукции: Предположим, что неравенство верно для некоторого натурального числа $k \ge 4$, то есть $3^k > 2^k + 7k$.
Докажем, что неравенство верно и для $n = k+1$: $3^{k+1} > 2^{k+1} + 7(k+1)$.
Используя предположение индукции, преобразуем левую часть:
$3^{k+1} = 3 \cdot 3^k > 3(2^k + 7k) = 3 \cdot 2^k + 21k$.
Правая часть неравенства для $k+1$ равна:
$2^{k+1} + 7(k+1) = 2 \cdot 2^k + 7k + 7$.
Теперь докажем, что $3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7$.
Для этого рассмотрим их разность:
$(3 \cdot 2^k + 21k) - (2 \cdot 2^k + 7k + 7) = (3 \cdot 2^k - 2 \cdot 2^k) + (21k - 7k) - 7 = 2^k + 14k - 7$.
Нам нужно показать, что $2^k + 14k - 7 > 0$ для всех $k \ge 4$.
При $k=4$: $2^4 + 14(4) - 7 = 16 + 56 - 7 = 65 > 0$.
Функция $f(k) = 2^k + 14k - 7$ является возрастающей для $k \ge 1$, так как оба слагаемых $2^k$ и $14k$ являются возрастающими функциями. Следовательно, если неравенство верно для $k=4$, оно будет верно и для всех $k > 4$.
Таким образом, мы доказали, что $3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7$.
Получаем цепочку неравенств: $3^{k+1} > 3 \cdot 2^k + 21k > 2 \cdot 2^k + 7k + 7 = 2^{k+1} + 7(k+1)$.
Шаг индукции доказан. По принципу математической индукции, неравенство $3^n > 2^n + 7n$ верно для всех натуральных чисел $n \ge 4$.
Ответ: для всех натуральных $n \ge 4$.
Другие задания:
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