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Номер 352, страница 106 - 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 352, страница 106.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108074 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "106" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-352" "field_display_title" => "352" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …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 {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …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 {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #original: array:24 [ "id" => 1107653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "92" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 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 {#1063} "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 {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1276707 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108074" "type" => "task" ] "text" => "<p><strong>352.</strong> Для последовательности $(b_n)$, заданной перечислением ее членов, докажите формулу суммы $n$ первых членов:</p><p><strong>a)</strong> $S_n = \frac{n(n+1)(2n+1)}{6}$, если $(b_n): 1^2; 2^2; 3^2; \ldots; n^2;$</p><p><strong>б)</strong> $S_n = \frac{n(4n^2-1)}{3}$, если $(b_n): 1^2; 3^2; \ldots; (2n-1)^2;$</p><p><strong>в)</strong> $S_n = \frac{2n(n+1)(2n+1)}{3}$, если $(b_n): 2^2; 4^2; \ldots; (2n)^2;$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "352-1.jpg" "alt" => null "width" => "1563" "height" => 608 "path" => "/media/algebra_09/soltan-u19/0-1/352-1.webp?ts=1752834398" ] ] ] #original: array:7 [ "id" => 1276707 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108074" "type" => "task" ] "text" => "<p><strong>352.</strong> Для последовательности $(b_n)$, заданной перечислением ее членов, докажите формулу суммы $n$ первых членов:</p><p><strong>a)</strong> $S_n = \frac{n(n+1)(2n+1)}{6}$, если $(b_n): 1^2; 2^2; 3^2; \ldots; n^2;$</p><p><strong>б)</strong> $S_n = \frac{n(4n^2-1)}{3}$, если $(b_n): 1^2; 3^2; \ldots; (2n-1)^2;$</p><p><strong>в)</strong> $S_n = \frac{2n(n+1)(2n+1)}{3}$, если $(b_n): 2^2; 4^2; \ldots; (2n)^2;$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "352-1.jpg" "alt" => null "width" => "1563" "height" => 608 "path" => "/media/algebra_09/soltan-u19/0-1/352-1.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 {#1107 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1277998 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108074" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "352-1.jpg" "alt" => null "width" => "1377" "height" => 2133 "path" => "/media/algebra_09/soltan-u19/1-1/352-1.webp?ts=1752831112" ] 1 => array:5 [ "name" => "352-2.jpg" "alt" => null "width" => "1410" "height" => 2398 "path" => "/media/algebra_09/soltan-u19/1-1/352-2.webp?ts=1752831112" ] ] ] #original: array:6 [ "id" => 1277998 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108074" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "352-1.jpg" "alt" => null "width" => "1377" "height" => 2133 "path" => "/media/algebra_09/soltan-u19/1-1/352-1.webp?ts=1752831112" ] 1 => array:5 [ "name" => "352-2.jpg" "alt" => null "width" => "1410" "height" => 2398 "path" => "/media/algebra_09/soltan-u19/1-1/352-2.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 {#1115 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1553953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108074" "type" => "task" ] "text" => "<p><strong>а)</strong> Докажем формулу $S_n = \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ методом математической индукции.</p><p><em>1. База индукции (проверка для n=1):</em></p><p>При $n=1$, левая часть формулы (сумма) равна $S_1 = 1^2 = 1$.</p><p>Правая часть формулы равна $\frac{1 \cdot (1+1) \cdot (2 \cdot 1+1)}{6} = \frac{1 \cdot 2 \cdot 3}{6} = 1$.</p><p>Так как $1 = 1$, формула верна для $n=1$.</p><p><em>2. Индукционное предположение (предположение для n=k):</em></p><p>Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:</p><p>$S_k = 1^2 + 2^2 + \dots + k^2 = \frac{k(k+1)(2k+1)}{6}$</p><p><em>3. Индукционный шаг (доказательство для n=k+1):</em></p><p>Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. То есть, нам нужно доказать, что $S_{k+1} = \frac{(k+1)((k+1)+1)(2(k+1)+1)}{6} = \frac{(k+1)(k+2)(2k+3)}{6}$.</p><p>Рассмотрим сумму $S_{k+1}$:</p><p>$S_{k+1} = 1^2 + 2^2 + \dots + k^2 + (k+1)^2 = S_k + (k+1)^2$.</p><p>Используя индукционное предположение для $S_k$, получаем:</p><p>$S_{k+1} = \frac{k(k+1)(2k+1)}{6} + (k+1)^2$.</p><p>Приведем к общему знаменателю и вынесем общий множитель $(k+1)$:</p><p>$S_{k+1} = (k+1) \left( \frac{k(2k+1)}{6} + k+1 \right) = (k+1) \left( \frac{k(2k+1) + 6(k+1)}{6} \right)$.</p><p>$S_{k+1} = (k+1) \left( \frac{2k^2+k+6k+6}{6} \right) = \frac{(k+1)(2k^2+7k+6)}{6}$.</p><p>Разложим квадратный трехчлен $2k^2+7k+6$ на множители: $2k^2+7k+6 = (k+2)(2k+3)$.</p><p>Подставим разложение в выражение для $S_{k+1}$:</p><p>$S_{k+1} = \frac{(k+1)(k+2)(2k+3)}{6}$.</p><p>Это и есть формула для $n=k+1$. Индукционный шаг доказан.</p><p>Таким образом, по принципу математической индукции, формула верна для любого натурального $n$.</p><p><strong>Ответ:</strong> Формула доказана.</p><p><strong>б)</strong> Докажем формулу $S_n = \sum_{k=1}^{n} (2k-1)^2 = \frac{n(4n^2-1)}{3}$ методом математической индукции.</p><p><em>1. База индукции (проверка для n=1):</em></p><p>При $n=1$, левая часть формулы (сумма) равна $S_1 = (2 \cdot 1-1)^2 = 1^2 = 1$.</p><p>Правая часть формулы равна $\frac{1 \cdot (4 \cdot 1^2 - 1)}{3} = \frac{1 \cdot 3}{3} = 1$.</p><p>Так как $1 = 1$, формула верна для $n=1$.</p><p><em>2. Индукционное предположение (предположение для n=k):</em></p><p>Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:</p><p>$S_k = 1^2 + 3^2 + \dots + (2k-1)^2 = \frac{k(4k^2-1)}{3}$.</p><p><em>3. Индукционный шаг (доказательство для n=k+1):</em></p><p>Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. Нам нужно доказать, что $S_{k+1} = \frac{(k+1)(4(k+1)^2-1)}{3}$.</p><p>Рассмотрим сумму $S_{k+1}$:</p><p>$S_{k+1} = 1^2 + 3^2 + \dots + (2k-1)^2 + (2(k+1)-1)^2 = S_k + (2k+1)^2$.</p><p>Используя индукционное предположение для $S_k$, получаем:</p><p>$S_{k+1} = \frac{k(4k^2-1)}{3} + (2k+1)^2$.</p><p>Используем формулу разности квадратов $4k^2-1 = (2k-1)(2k+1)$:</p><p>$S_{k+1} = \frac{k(2k-1)(2k+1)}{3} + (2k+1)^2$.</p><p>Вынесем общий множитель $(2k+1)$:</p><p>$S_{k+1} = (2k+1) \left( \frac{k(2k-1)}{3} + (2k+1) \right) = (2k+1) \left( \frac{2k^2-k + 3(2k+1)}{3} \right)$.</p><p>$S_{k+1} = (2k+1) \frac{2k^2-k+6k+3}{3} = \frac{(2k+1)(2k^2+5k+3)}{3}$.</p><p>Разложим квадратный трехчлен $2k^2+5k+3$ на множители: $2k^2+5k+3 = (k+1)(2k+3)$.</p><p>Подставим разложение в выражение для $S_{k+1}$:</p><p>$S_{k+1} = \frac{(k+1)(2k+1)(2k+3)}{3}$.</p><p>Теперь преобразуем правую часть целевой формулы для $n=k+1$: $\frac{(k+1)(4(k+1)^2-1)}{3}$.</p><p>$4(k+1)^2-1 = (2(k+1))^2-1^2 = (2(k+1)-1)(2(k+1)+1) = (2k+1)(2k+3)$.</p><p>Значит, правая часть равна $\frac{(k+1)(2k+1)(2k+3)}{3}$, что совпадает с нашим результатом. Индукционный шаг доказан.</p><p>Таким образом, по принципу математической индукции, формула верна для любого натурального $n$.</p><p><strong>Ответ:</strong> Формула доказана.</p><p><strong>в)</strong> Докажем формулу $S_n = \sum_{k=1}^{n} (2k)^2 = \frac{2n(n+1)(2n+1)}{3}$.</p><p><em>Способ 1: Использование результата из пункта а).</em></p><p>Сумма квадратов первых $n$ четных чисел:</p><p>$S_n = 2^2 + 4^2 + \dots + (2n)^2 = \sum_{k=1}^{n} (2k)^2 = \sum_{k=1}^{n} 4k^2$.</p><p>Вынесем константу 4 за знак суммы:</p><p>$S_n = 4 \sum_{k=1}^{n} k^2 = 4(1^2+2^2+\dots+n^2)$.</p><p>Из пункта а) известно, что $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.</p><p>Подставим эту формулу:</p><p>$S_n = 4 \cdot \frac{n(n+1)(2n+1)}{6} = \frac{2n(n+1)(2n+1)}{3}$.</p><p><em>Способ 2: Метод математической индукции.</em></p><p><em>1. База индукции (проверка для n=1):</em></p><p>При $n=1$, левая часть формулы (сумма) равна $S_1 = (2 \cdot 1)^2 = 4$.</p><p>Правая часть формулы равна $\frac{2 \cdot 1 \cdot (1+1) \cdot (2 \cdot 1+1)}{3} = \frac{2 \cdot 2 \cdot 3}{3} = 4$.</p><p>Так как $4 = 4$, формула верна для $n=1$.</p><p><em>2. Индукционное предположение (предположение для n=k):</em></p><p>Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:</p><p>$S_k = 2^2 + 4^2 + \dots + (2k)^2 = \frac{2k(k+1)(2k+1)}{3}$.</p><p><em>3. Индукционный шаг (доказательство для n=k+1):</em></p><p>Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. Нам нужно доказать, что $S_{k+1} = \frac{2(k+1)(k+2)(2k+3)}{3}$.</p><p>Рассмотрим сумму $S_{k+1}$:</p><p>$S_{k+1} = 2^2 + 4^2 + \dots + (2k)^2 + (2(k+1))^2 = S_k + 4(k+1)^2$.</p><p>Используя индукционное предположение для $S_k$, получаем:</p><p>$S_{k+1} = \frac{2k(k+1)(2k+1)}{3} + 4(k+1)^2$.</p><p>Вынесем общий множитель $2(k+1)$:</p><p>$S_{k+1} = 2(k+1) \left( \frac{k(2k+1)}{3} + 2(k+1) \right) = 2(k+1) \left( \frac{2k^2+k + 6(k+1)}{3} \right)$.</p><p>$S_{k+1} = 2(k+1) \left( \frac{2k^2+k+6k+6}{3} \right) = \frac{2(k+1)(2k^2+7k+6)}{3}$.</p><p>Как мы уже находили в пункте а), $2k^2+7k+6 = (k+2)(2k+3)$.</p><p>Подставим разложение в выражение для $S_{k+1}$:</p><p>$S_{k+1} = \frac{2(k+1)(k+2)(2k+3)}{3}$.</p><p>Это и есть формула для $n=k+1$. Индукционный шаг доказан.</p><p>Таким образом, обоими способами мы доказали, что формула верна для любого натурального $n$.</p><p><strong>Ответ:</strong> Формула доказана.</p>" ] #original: array:6 [ "id" => 1553953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108074" "type" => "task" ] "text" => "<p><strong>а)</strong> Докажем формулу $S_n = \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ методом математической индукции.</p><p><em>1. База индукции (проверка для n=1):</em></p><p>При $n=1$, левая часть формулы (сумма) равна $S_1 = 1^2 = 1$.</p><p>Правая часть формулы равна $\frac{1 \cdot (1+1) \cdot (2 \cdot 1+1)}{6} = \frac{1 \cdot 2 \cdot 3}{6} = 1$.</p><p>Так как $1 = 1$, формула верна для $n=1$.</p><p><em>2. Индукционное предположение (предположение для n=k):</em></p><p>Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:</p><p>$S_k = 1^2 + 2^2 + \dots + k^2 = \frac{k(k+1)(2k+1)}{6}$</p><p><em>3. Индукционный шаг (доказательство для n=k+1):</em></p><p>Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. То есть, нам нужно доказать, что $S_{k+1} = \frac{(k+1)((k+1)+1)(2(k+1)+1)}{6} = \frac{(k+1)(k+2)(2k+3)}{6}$.</p><p>Рассмотрим сумму $S_{k+1}$:</p><p>$S_{k+1} = 1^2 + 2^2 + \dots + k^2 + (k+1)^2 = S_k + (k+1)^2$.</p><p>Используя индукционное предположение для $S_k$, получаем:</p><p>$S_{k+1} = \frac{k(k+1)(2k+1)}{6} + (k+1)^2$.</p><p>Приведем к общему знаменателю и вынесем общий множитель $(k+1)$:</p><p>$S_{k+1} = (k+1) \left( \frac{k(2k+1)}{6} + k+1 \right) = (k+1) \left( \frac{k(2k+1) + 6(k+1)}{6} \right)$.</p><p>$S_{k+1} = (k+1) \left( \frac{2k^2+k+6k+6}{6} \right) = \frac{(k+1)(2k^2+7k+6)}{6}$.</p><p>Разложим квадратный трехчлен $2k^2+7k+6$ на множители: $2k^2+7k+6 = (k+2)(2k+3)$.</p><p>Подставим разложение в выражение для $S_{k+1}$:</p><p>$S_{k+1} = \frac{(k+1)(k+2)(2k+3)}{6}$.</p><p>Это и есть формула для $n=k+1$. Индукционный шаг доказан.</p><p>Таким образом, по принципу математической индукции, формула верна для любого натурального $n$.</p><p><strong>Ответ:</strong> Формула доказана.</p><p><strong>б)</strong> Докажем формулу $S_n = \sum_{k=1}^{n} (2k-1)^2 = \frac{n(4n^2-1)}{3}$ методом математической индукции.</p><p><em>1. База индукции (проверка для n=1):</em></p><p>При $n=1$, левая часть формулы (сумма) равна $S_1 = (2 \cdot 1-1)^2 = 1^2 = 1$.</p><p>Правая часть формулы равна $\frac{1 \cdot (4 \cdot 1^2 - 1)}{3} = \frac{1 \cdot 3}{3} = 1$.</p><p>Так как $1 = 1$, формула верна для $n=1$.</p><p><em>2. Индукционное предположение (предположение для n=k):</em></p><p>Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:</p><p>$S_k = 1^2 + 3^2 + \dots + (2k-1)^2 = \frac{k(4k^2-1)}{3}$.</p><p><em>3. Индукционный шаг (доказательство для n=k+1):</em></p><p>Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. Нам нужно доказать, что $S_{k+1} = \frac{(k+1)(4(k+1)^2-1)}{3}$.</p><p>Рассмотрим сумму $S_{k+1}$:</p><p>$S_{k+1} = 1^2 + 3^2 + \dots + (2k-1)^2 + (2(k+1)-1)^2 = S_k + (2k+1)^2$.</p><p>Используя индукционное предположение для $S_k$, получаем:</p><p>$S_{k+1} = \frac{k(4k^2-1)}{3} + (2k+1)^2$.</p><p>Используем формулу разности квадратов $4k^2-1 = (2k-1)(2k+1)$:</p><p>$S_{k+1} = \frac{k(2k-1)(2k+1)}{3} + (2k+1)^2$.</p><p>Вынесем общий множитель $(2k+1)$:</p><p>$S_{k+1} = (2k+1) \left( \frac{k(2k-1)}{3} + (2k+1) \right) = (2k+1) \left( \frac{2k^2-k + 3(2k+1)}{3} \right)$.</p><p>$S_{k+1} = (2k+1) \frac{2k^2-k+6k+3}{3} = \frac{(2k+1)(2k^2+5k+3)}{3}$.</p><p>Разложим квадратный трехчлен $2k^2+5k+3$ на множители: $2k^2+5k+3 = (k+1)(2k+3)$.</p><p>Подставим разложение в выражение для $S_{k+1}$:</p><p>$S_{k+1} = \frac{(k+1)(2k+1)(2k+3)}{3}$.</p><p>Теперь преобразуем правую часть целевой формулы для $n=k+1$: $\frac{(k+1)(4(k+1)^2-1)}{3}$.</p><p>$4(k+1)^2-1 = (2(k+1))^2-1^2 = (2(k+1)-1)(2(k+1)+1) = (2k+1)(2k+3)$.</p><p>Значит, правая часть равна $\frac{(k+1)(2k+1)(2k+3)}{3}$, что совпадает с нашим результатом. Индукционный шаг доказан.</p><p>Таким образом, по принципу математической индукции, формула верна для любого натурального $n$.</p><p><strong>Ответ:</strong> Формула доказана.</p><p><strong>в)</strong> Докажем формулу $S_n = \sum_{k=1}^{n} (2k)^2 = \frac{2n(n+1)(2n+1)}{3}$.</p><p><em>Способ 1: Использование результата из пункта а).</em></p><p>Сумма квадратов первых $n$ четных чисел:</p><p>$S_n = 2^2 + 4^2 + \dots + (2n)^2 = \sum_{k=1}^{n} (2k)^2 = \sum_{k=1}^{n} 4k^2$.</p><p>Вынесем константу 4 за знак суммы:</p><p>$S_n = 4 \sum_{k=1}^{n} k^2 = 4(1^2+2^2+\dots+n^2)$.</p><p>Из пункта а) известно, что $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.</p><p>Подставим эту формулу:</p><p>$S_n = 4 \cdot \frac{n(n+1)(2n+1)}{6} = \frac{2n(n+1)(2n+1)}{3}$.</p><p><em>Способ 2: Метод математической индукции.</em></p><p><em>1. База индукции (проверка для n=1):</em></p><p>При $n=1$, левая часть формулы (сумма) равна $S_1 = (2 \cdot 1)^2 = 4$.</p><p>Правая часть формулы равна $\frac{2 \cdot 1 \cdot (1+1) \cdot (2 \cdot 1+1)}{3} = \frac{2 \cdot 2 \cdot 3}{3} = 4$.</p><p>Так как $4 = 4$, формула верна для $n=1$.</p><p><em>2. Индукционное предположение (предположение для n=k):</em></p><p>Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:</p><p>$S_k = 2^2 + 4^2 + \dots + (2k)^2 = \frac{2k(k+1)(2k+1)}{3}$.</p><p><em>3. Индукционный шаг (доказательство для n=k+1):</em></p><p>Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. Нам нужно доказать, что $S_{k+1} = \frac{2(k+1)(k+2)(2k+3)}{3}$.</p><p>Рассмотрим сумму $S_{k+1}$:</p><p>$S_{k+1} = 2^2 + 4^2 + \dots + (2k)^2 + (2(k+1))^2 = S_k + 4(k+1)^2$.</p><p>Используя индукционное предположение для $S_k$, получаем:</p><p>$S_{k+1} = \frac{2k(k+1)(2k+1)}{3} + 4(k+1)^2$.</p><p>Вынесем общий множитель $2(k+1)$:</p><p>$S_{k+1} = 2(k+1) \left( \frac{k(2k+1)}{3} + 2(k+1) \right) = 2(k+1) \left( \frac{2k^2+k + 6(k+1)}{3} \right)$.</p><p>$S_{k+1} = 2(k+1) \left( \frac{2k^2+k+6k+6}{3} \right) = \frac{2(k+1)(2k^2+7k+6)}{3}$.</p><p>Как мы уже находили в пункте а), $2k^2+7k+6 = (k+2)(2k+3)$.</p><p>Подставим разложение в выражение для $S_{k+1}$:</p><p>$S_{k+1} = \frac{2(k+1)(k+2)(2k+3)}{3}$.</p><p>Это и есть формула для $n=k+1$. Индукционный шаг доказан.</p><p>Таким образом, обоими способами мы доказали, что формула верна для любого натурального $n$.</p><p><strong>Ответ:</strong> Формула доказана.</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" => "1108075" "type" => "task" ] "previous" => array:2 [ "refs" => "1108073" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 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"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 {#1135 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Term {#1137 #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 {#1138 #items: array:1 [ 0 => App\Models\Term {#1139 #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 {#1140 #items: array:1 [ 0 => App\Models\Term {#1141 #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 {#1142 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147 #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 {#1148 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151 #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 {#1125} "field_class" => Illuminate\Database\Eloquent\Collection {#1127} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129} "field_author" => Illuminate\Database\Eloquent\Collection {#1131} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1135} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_country" => Illuminate\Database\Eloquent\Collection {#1138} "field_city" => Illuminate\Database\Eloquent\Collection {#1140} "field_series" => Illuminate\Database\Eloquent\Collection {#1142} "field_umk" => Illuminate\Database\Eloquent\Collection {#1143} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "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 {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1166 #items: array:1 [ 0 => App\Models\BookPage {#1165 #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" => 1097848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "106" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-106" "field_display_title" => "106" "field_folder" => "folder1" "field_image_name" => "106" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Book {#1168 #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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1196 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:1 [ 0 => App\Models\Element {#1206 #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" => 1260999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 …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" => "1097849" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097847" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1337 #items: array:9 [ 0 => App\Models\Task {#1353 #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" => 1108074 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "106" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-352" "field_display_title" => "352" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1357 …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 {#1356 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1391 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1458 …2} "content" => Illuminate\Database\Eloquent\Collection {#1484 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1485 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108074 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "106" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-352" "field_display_title" => "352" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1357 …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 {#1356 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1391 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1458 …2} "content" => Illuminate\Database\Eloquent\Collection {#1484 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1485 …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 {#1483 #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" => 1108075 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "106" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-353" "field_display_title" => "353" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1486 …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 {#1487 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1488 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1489 …2} "content" => Illuminate\Database\Eloquent\Collection {#1499 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1506 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108075 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "106" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-353" "field_display_title" => "353" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1486 …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 {#1487 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1488 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1489 …2} "content" => Illuminate\Database\Eloquent\Collection {#1499 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1506 …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 {#1490 #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" => 1108076 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "106" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-354" "field_display_title" => "354" "field_outside_task" => 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№352 (с. 106)
Условие. №352 (с. 106)
скриншот условия
352. Для последовательности $(b_n)$, заданной перечислением ее членов, докажите формулу суммы $n$ первых членов:
a) $S_n = \frac{n(n+1)(2n+1)}{6}$, если $(b_n): 1^2; 2^2; 3^2; \ldots; n^2;$
б) $S_n = \frac{n(4n^2-1)}{3}$, если $(b_n): 1^2; 3^2; \ldots; (2n-1)^2;$
в) $S_n = \frac{2n(n+1)(2n+1)}{3}$, если $(b_n): 2^2; 4^2; \ldots; (2n)^2;$
Решение. №352 (с. 106)
Решение 2 (rus). №352 (с. 106)
а) Докажем формулу $S_n = \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ методом математической индукции.
1. База индукции (проверка для n=1):
При $n=1$, левая часть формулы (сумма) равна $S_1 = 1^2 = 1$.
Правая часть формулы равна $\frac{1 \cdot (1+1) \cdot (2 \cdot 1+1)}{6} = \frac{1 \cdot 2 \cdot 3}{6} = 1$.
Так как $1 = 1$, формула верна для $n=1$.
2. Индукционное предположение (предположение для n=k):
Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:
$S_k = 1^2 + 2^2 + \dots + k^2 = \frac{k(k+1)(2k+1)}{6}$
3. Индукционный шаг (доказательство для n=k+1):
Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. То есть, нам нужно доказать, что $S_{k+1} = \frac{(k+1)((k+1)+1)(2(k+1)+1)}{6} = \frac{(k+1)(k+2)(2k+3)}{6}$.
Рассмотрим сумму $S_{k+1}$:
$S_{k+1} = 1^2 + 2^2 + \dots + k^2 + (k+1)^2 = S_k + (k+1)^2$.
Используя индукционное предположение для $S_k$, получаем:
$S_{k+1} = \frac{k(k+1)(2k+1)}{6} + (k+1)^2$.
Приведем к общему знаменателю и вынесем общий множитель $(k+1)$:
$S_{k+1} = (k+1) \left( \frac{k(2k+1)}{6} + k+1 \right) = (k+1) \left( \frac{k(2k+1) + 6(k+1)}{6} \right)$.
$S_{k+1} = (k+1) \left( \frac{2k^2+k+6k+6}{6} \right) = \frac{(k+1)(2k^2+7k+6)}{6}$.
Разложим квадратный трехчлен $2k^2+7k+6$ на множители: $2k^2+7k+6 = (k+2)(2k+3)$.
Подставим разложение в выражение для $S_{k+1}$:
$S_{k+1} = \frac{(k+1)(k+2)(2k+3)}{6}$.
Это и есть формула для $n=k+1$. Индукционный шаг доказан.
Таким образом, по принципу математической индукции, формула верна для любого натурального $n$.
Ответ: Формула доказана.
б) Докажем формулу $S_n = \sum_{k=1}^{n} (2k-1)^2 = \frac{n(4n^2-1)}{3}$ методом математической индукции.
1. База индукции (проверка для n=1):
При $n=1$, левая часть формулы (сумма) равна $S_1 = (2 \cdot 1-1)^2 = 1^2 = 1$.
Правая часть формулы равна $\frac{1 \cdot (4 \cdot 1^2 - 1)}{3} = \frac{1 \cdot 3}{3} = 1$.
Так как $1 = 1$, формула верна для $n=1$.
2. Индукционное предположение (предположение для n=k):
Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:
$S_k = 1^2 + 3^2 + \dots + (2k-1)^2 = \frac{k(4k^2-1)}{3}$.
3. Индукционный шаг (доказательство для n=k+1):
Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. Нам нужно доказать, что $S_{k+1} = \frac{(k+1)(4(k+1)^2-1)}{3}$.
Рассмотрим сумму $S_{k+1}$:
$S_{k+1} = 1^2 + 3^2 + \dots + (2k-1)^2 + (2(k+1)-1)^2 = S_k + (2k+1)^2$.
Используя индукционное предположение для $S_k$, получаем:
$S_{k+1} = \frac{k(4k^2-1)}{3} + (2k+1)^2$.
Используем формулу разности квадратов $4k^2-1 = (2k-1)(2k+1)$:
$S_{k+1} = \frac{k(2k-1)(2k+1)}{3} + (2k+1)^2$.
Вынесем общий множитель $(2k+1)$:
$S_{k+1} = (2k+1) \left( \frac{k(2k-1)}{3} + (2k+1) \right) = (2k+1) \left( \frac{2k^2-k + 3(2k+1)}{3} \right)$.
$S_{k+1} = (2k+1) \frac{2k^2-k+6k+3}{3} = \frac{(2k+1)(2k^2+5k+3)}{3}$.
Разложим квадратный трехчлен $2k^2+5k+3$ на множители: $2k^2+5k+3 = (k+1)(2k+3)$.
Подставим разложение в выражение для $S_{k+1}$:
$S_{k+1} = \frac{(k+1)(2k+1)(2k+3)}{3}$.
Теперь преобразуем правую часть целевой формулы для $n=k+1$: $\frac{(k+1)(4(k+1)^2-1)}{3}$.
$4(k+1)^2-1 = (2(k+1))^2-1^2 = (2(k+1)-1)(2(k+1)+1) = (2k+1)(2k+3)$.
Значит, правая часть равна $\frac{(k+1)(2k+1)(2k+3)}{3}$, что совпадает с нашим результатом. Индукционный шаг доказан.
Таким образом, по принципу математической индукции, формула верна для любого натурального $n$.
Ответ: Формула доказана.
в) Докажем формулу $S_n = \sum_{k=1}^{n} (2k)^2 = \frac{2n(n+1)(2n+1)}{3}$.
Способ 1: Использование результата из пункта а).
Сумма квадратов первых $n$ четных чисел:
$S_n = 2^2 + 4^2 + \dots + (2n)^2 = \sum_{k=1}^{n} (2k)^2 = \sum_{k=1}^{n} 4k^2$.
Вынесем константу 4 за знак суммы:
$S_n = 4 \sum_{k=1}^{n} k^2 = 4(1^2+2^2+\dots+n^2)$.
Из пункта а) известно, что $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.
Подставим эту формулу:
$S_n = 4 \cdot \frac{n(n+1)(2n+1)}{6} = \frac{2n(n+1)(2n+1)}{3}$.
Способ 2: Метод математической индукции.
1. База индукции (проверка для n=1):
При $n=1$, левая часть формулы (сумма) равна $S_1 = (2 \cdot 1)^2 = 4$.
Правая часть формулы равна $\frac{2 \cdot 1 \cdot (1+1) \cdot (2 \cdot 1+1)}{3} = \frac{2 \cdot 2 \cdot 3}{3} = 4$.
Так как $4 = 4$, формула верна для $n=1$.
2. Индукционное предположение (предположение для n=k):
Предположим, что формула верна для некоторого натурального числа $n=k$, то есть:
$S_k = 2^2 + 4^2 + \dots + (2k)^2 = \frac{2k(k+1)(2k+1)}{3}$.
3. Индукционный шаг (доказательство для n=k+1):
Докажем, что если формула верна для $n=k$, то она верна и для $n=k+1$. Нам нужно доказать, что $S_{k+1} = \frac{2(k+1)(k+2)(2k+3)}{3}$.
Рассмотрим сумму $S_{k+1}$:
$S_{k+1} = 2^2 + 4^2 + \dots + (2k)^2 + (2(k+1))^2 = S_k + 4(k+1)^2$.
Используя индукционное предположение для $S_k$, получаем:
$S_{k+1} = \frac{2k(k+1)(2k+1)}{3} + 4(k+1)^2$.
Вынесем общий множитель $2(k+1)$:
$S_{k+1} = 2(k+1) \left( \frac{k(2k+1)}{3} + 2(k+1) \right) = 2(k+1) \left( \frac{2k^2+k + 6(k+1)}{3} \right)$.
$S_{k+1} = 2(k+1) \left( \frac{2k^2+k+6k+6}{3} \right) = \frac{2(k+1)(2k^2+7k+6)}{3}$.
Как мы уже находили в пункте а), $2k^2+7k+6 = (k+2)(2k+3)$.
Подставим разложение в выражение для $S_{k+1}$:
$S_{k+1} = \frac{2(k+1)(k+2)(2k+3)}{3}$.
Это и есть формула для $n=k+1$. Индукционный шаг доказан.
Таким образом, обоими способами мы доказали, что формула верна для любого натурального $n$.
Ответ: Формула доказана.
Другие задания:
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