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Номер 357, страница 133 - гдз по алгебре 9 класс учебник Бунимович, Кузнецова
Авторы: Бунимович Е. А., Кузнецова Л. В., Минаева С. С., Рослова Л. О., Суворова С. Б.
Тип: Учебник
Серия: сферы 1-11
Издательство: Просвещение
Год издания: 2018 - 2026
Цвет обложки: оранжевый с диаграммой
ISBN: 978-5-09-051312-8
Популярные ГДЗ в 9 классе
Глава 4. Арифметическая и геометрическая прогрессия. 4.1 Числовые последовательности - номер 357, страница 133.
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" => 199133 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "133" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-357" "field_display_title" => "357" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 198683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Арифметическая и геометрическая прогрессия" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "127" "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 {#1083 #items: array:1 [ 0 => App\Models\Book {#1049 #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" => 2789 "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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #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 {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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 {#1056 #items: array:5 [ 0 => App\Models\Term {#1057 #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" => 3941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бунимович" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Евгений" "field_patronymic" => "Абрамович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бунимович" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Евгений" "field_patronymic" => "Абрамович" "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 {#1058 #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" => 4848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кузнецова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 4848 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кузнецова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Викторовна" "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 {#1059 #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" => 5198 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Минаева" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Станиславовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5198 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Минаева" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Станиславовна" "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 => "*" ] } 3 => App\Models\Term {#1060 #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" => 5622 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Рослова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лариса" "field_patronymic" => "Олеговна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5622 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Рослова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лариса" "field_patronymic" => "Олеговна" "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 => "*" ] } 4 => App\Models\Term {#1061 #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" => 5879 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Суворова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Борисовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 5879 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Суворова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Светлана" "field_patronymic" => "Борисовна" "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 {#1062 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #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" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "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" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "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" => "rossiya" ] #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 {#1067 #items: array:1 [ 0 => App\Models\Term {#1068 #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" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "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" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "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" => "moskva" ] #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 {#1069 #items: array:1 [ 0 => App\Models\Term {#1070 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false 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Алгебраические выражения порой способны поставить в тупик и отличников, что уж говорить о подростках, которые больше склонны к гуманитарным дисциплинам? Поэтому помощь <strong>ГДЗ по алгебре 9 класс учебное пособие Бунимович, Кузнецова, Минаева (Просвещение)</strong> может оказаться незаменимой, особенно при текущих нагрузках. </p> <p> Больше всего трудностей возникает у ребят при изучении следующих тем: </p> <ol> <li>Общие свойства неравенств.</li> <li>Квадратичная функция и её график.</li> <li>Рациональные выражения.</li> <li>Числовые последовательности.</li> <li>Интервальный ряд. Гистограмма.</li> <li>Статистическое оценивание и прогноз.</li> </ol> <p> Уроки нередко проходят срезано, так как за 45 минут преподавателям нужно многое успеть. Они объясняют лишь часть темы, оставляя все остальное для самостоятельного разбора. Но много ли учеников проводит дома свободное время с учебником? Конечно, нет. В основном учащиеся наспех выполняют заданные номера и моментально забывают о них. 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Поэтому большинство школьников подходит к работе с <strong>решебником Бунимовича по алгебре за 9 класс</strong> ответственно и внимательно. </p> <p> Учителя-предметники, да и сами авторы пособия рекомендуют всегда придерживаться следующих правил: </p> <ol> <li>не забывать про освоение теоретического материала;</li> <li>выполнять домашние задания без подсказок со стороны;</li> <li>когда все сделано, можно приступать к сверке решений по ГДЗ;</li> <li>работа над ошибками - тоже немаловажный аспект.</li> </ol> <p> Подобный подход несомненно принесет пользу, а результаты станут заметны через пару занятий. Первая же контрольная принесет хорошие плоды, так как перед этим подростки подробно проработали все аспекты каждого параграфа. Происходит это ненавязчиво и не отнимает много времени, что придется по душе всем ребятам. </p> <h2> Чем именно полезен решебник? </h2> <p> Современная учебная программа в основном направлена на практику. Считается, что чем больше упражнений выполнят школьники, тем лучше они поймут новую тему. Но не всегда такой метод срабатывает. Все же понимание теории - немаловажная составляющая полноценных знаний по предмету. Но в условиях тотальных нагрузок преподавателям просто не остается достаточно времени для проработки всех нюансов. Так что у ребят остается один шанс не упустить ничего важного - это использовать <strong>гдз по алгебре 9 класс к учебнику Бунимович Е.А (2018-2023 гг.)</strong>. </p> <p> Для того чтобы упростить ученикам понимание параграфов, авторы издания разместили в нем: </p> <ul> <li>подробные решения ко всем номерам;</li> <li>верные ответы;</li> <li>готовые графики и заполненные таблицы;</li> <li>детальные наглядные примеры.</li> </ul> <p> Все, что нужно сделать подросткам - это правильно воспользоваться предоставленными сведениями. Расположение решебника и постоянный онлайн-доступ делают работу с ним простым делом, которое займет от силы несколько минут. 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#primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 198683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Арифметическая и геометрическая прогрессия" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "127" "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 {#1125 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1126 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 198683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Арифметическая и геометрическая прогрессия" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1124} "field_page_start" => "127" "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 {#1125} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1126} ] #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 {#1127 #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" => 198684 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4.1 Числовые последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "128" "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 {#1129 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 198684 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4.1 Числовые последовательности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1128} "field_page_start" => "128" "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 {#1129} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1130} ] #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 {#1112 #items: array:3 [ 0 => App\Models\Element {#1105 #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" => 210803 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1104 #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" => 700 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 #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" => "tolik" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1100 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 700 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102} "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" => "tolik" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1100} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1101} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098} "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" => "199133" "type" => "task" ] "text" => "<p><strong>357 1)</strong> Члены последовательности можно изображать точками на координатной плоскости: каждому члену соответствует точка, абсцисса которой равна его номеру, а ордината — значению этого члена. На рисунке <em>4.1</em> на координатной плоскости изображены первые шесть членов последовательности Фибоначчи: 1; 1; 2; 3; 5; 8. Им соответствуют точки: (1; 1), (2; 1), (3; 2), (4; 3), (5; 5), (6; 8). Назовите координаты точек, изображающих седьмой и восьмой члены этой последовательности.</p><p><strong>2)</strong> Изобразите точками на координатной плоскости первые шесть членов последовательности $(a_n)$, если:</p><p><strong>а)</strong> $a_n = n - 4$;</p><p><strong>в)</strong> $a_n = \frac{12}{n}$;</p><p><strong>б)</strong> $a_n = \frac{n}{2}$;</p><p><strong>г)</strong> $a_n = 0,5n^2 - 8$.</p><p>В каждом случае скажите, на какой линии располагаются построенные точки; проведите тонко эту линию.</p>" "img" => array:3 [ 0 => array:5 [ "name" => "357-1.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/0-01/357-1.webp?ts=1734190387" ] 1 => array:5 [ "name" => "357-2.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/0-01/357-2.webp?ts=1734190387" ] 2 => array:5 [ "name" => "357-3.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/0-01/357-3.webp?ts=1734190387" ] ] ] #original: array:7 [ "id" => 210803 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "199133" "type" => "task" ] "text" => "<p><strong>357 1)</strong> Члены последовательности можно изображать точками на координатной плоскости: каждому члену соответствует точка, абсцисса которой равна его номеру, а ордината — значению этого члена. На рисунке <em>4.1</em> на координатной плоскости изображены первые шесть членов последовательности Фибоначчи: 1; 1; 2; 3; 5; 8. Им соответствуют точки: (1; 1), (2; 1), (3; 2), (4; 3), (5; 5), (6; 8). Назовите координаты точек, изображающих седьмой и восьмой члены этой последовательности.</p><p><strong>2)</strong> Изобразите точками на координатной плоскости первые шесть членов последовательности $(a_n)$, если:</p><p><strong>а)</strong> $a_n = n - 4$;</p><p><strong>в)</strong> $a_n = \frac{12}{n}$;</p><p><strong>б)</strong> $a_n = \frac{n}{2}$;</p><p><strong>г)</strong> $a_n = 0,5n^2 - 8$.</p><p>В каждом случае скажите, на какой линии располагаются построенные точки; проведите тонко эту линию.</p>" "img" => array:3 [ 0 => array:5 [ "name" => "357-1.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/0-01/357-1.webp?ts=1734190387" ] 1 => array:5 [ "name" => "357-2.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/0-01/357-2.webp?ts=1734190387" ] 2 => array:5 [ "name" => "357-3.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/0-01/357-3.webp?ts=1734190387" ] ] ] #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 {#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:6 [ "id" => 211949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1096 #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" => 701 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 #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" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 701 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095} "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" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092} "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" => "199133" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "357-1.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/1-01/357-1.webp?ts=1734190447" ] ] ] #original: array:6 [ "id" => 211949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "199133" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "357-1.jpg" "alt" => null "width" => null "height" => null "path" => "/media/algebra_09/bunimovich-u18/1-01/357-1.webp?ts=1734190447" ] ] ] #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 {#1091 #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" => 1421615 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:1 [ 0 => App\Models\Edition {#1090 #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" => 5321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 #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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085} "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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084} "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" => "199133" "type" => "task" ] "text" => "<p><strong>1)</strong> Последовательность Фибоначчи задается рекуррентной формулой $a_1 = 1$, $a_2 = 1$, $a_n = a_{n-1} + a_{n-2}$ для $n > 2$. Первые шесть членов последовательности: 1, 1, 2, 3, 5, 8. То есть, $a_1=1, a_2=1, a_3=2, a_4=3, a_5=5, a_6=8$.</p><p>Чтобы найти седьмой член последовательности, нужно сложить пятый и шестой члены:</p><p>$a_7 = a_5 + a_6 = 5 + 8 = 13$.</p><p>Координаты точки, изображающей седьмой член, равны номеру члена (абсцисса) и значению члена (ордината), то есть (7; 13).</p><p>Чтобы найти восьмой член последовательности, нужно сложить шестой и седьмой члены:</p><p>$a_8 = a_6 + a_7 = 8 + 13 = 21$.</p><p>Координаты точки, изображающей восьмой член, равны (8; 21).</p><p>Ответ: координаты седьмой точки (7; 13), восьмой точки (8; 21).</p><p><strong>2)</strong></p><p><strong>а)</strong> Для последовательности $a_n = n - 4$ найдем первые шесть членов:</p><p>$a_1 = 1 - 4 = -3$</p><p>$a_2 = 2 - 4 = -2$</p><p>$a_3 = 3 - 4 = -1$</p><p>$a_4 = 4 - 4 = 0$</p><p>$a_5 = 5 - 4 = 1$</p><p>$a_6 = 6 - 4 = 2$</p><p>Координаты соответствующих точек: (1; -3), (2; -2), (3; -1), (4; 0), (5; 1), (6; 2). Эти точки лежат на прямой, заданной уравнением $y = x - 4$.</p><p>Ответ: точки (1; -3), (2; -2), (3; -1), (4; 0), (5; 1), (6; 2) лежат на прямой $y = x - 4$.</p><p><strong>б)</strong> Для последовательности $a_n = \frac{n}{2}$ найдем первые шесть членов:</p><p>$a_1 = \frac{1}{2} = 0,5$</p><p>$a_2 = \frac{2}{2} = 1$</p><p>$a_3 = \frac{3}{2} = 1,5$</p><p>$a_4 = \frac{4}{2} = 2$</p><p>$a_5 = \frac{5}{2} = 2,5$</p><p>$a_6 = \frac{6}{2} = 3$</p><p>Координаты соответствующих точек: (1; 0,5), (2; 1), (3; 1,5), (4; 2), (5; 2,5), (6; 3). Эти точки лежат на прямой, заданной уравнением $y = \frac{x}{2}$ или $y = 0,5x$.</p><p>Ответ: точки (1; 0,5), (2; 1), (3; 1,5), (4; 2), (5; 2,5), (6; 3) лежат на прямой $y = \frac{x}{2}$.</p><p><strong>в)</strong> Для последовательности $a_n = \frac{12}{n}$ найдем первые шесть членов:</p><p>$a_1 = \frac{12}{1} = 12$</p><p>$a_2 = \frac{12}{2} = 6$</p><p>$a_3 = \frac{12}{3} = 4$</p><p>$a_4 = \frac{12}{4} = 3$</p><p>$a_5 = \frac{12}{5} = 2,4$</p><p>$a_6 = \frac{12}{6} = 2$</p><p>Координаты соответствующих точек: (1; 12), (2; 6), (3; 4), (4; 3), (5; 2,4), (6; 2). Эти точки лежат на кривой, которая является ветвью гиперболы, заданной уравнением $y = \frac{12}{x}$.</p><p>Ответ: точки (1; 12), (2; 6), (3; 4), (4; 3), (5; 2,4), (6; 2) лежат на гиперболе $y = \frac{12}{x}$.</p><p><strong>г)</strong> Для последовательности $a_n = 0,5n^2 - 8$ найдем первые шесть членов:</p><p>$a_1 = 0,5 \cdot 1^2 - 8 = 0,5 - 8 = -7,5$</p><p>$a_2 = 0,5 \cdot 2^2 - 8 = 0,5 \cdot 4 - 8 = 2 - 8 = -6$</p><p>$a_3 = 0,5 \cdot 3^2 - 8 = 0,5 \cdot 9 - 8 = 4,5 - 8 = -3,5$</p><p>$a_4 = 0,5 \cdot 4^2 - 8 = 0,5 \cdot 16 - 8 = 8 - 8 = 0$</p><p>$a_5 = 0,5 \cdot 5^2 - 8 = 0,5 \cdot 25 - 8 = 12,5 - 8 = 4,5$</p><p>$a_6 = 0,5 \cdot 6^2 - 8 = 0,5 \cdot 36 - 8 = 18 - 8 = 10$</p><p>Координаты соответствующих точек: (1; -7,5), (2; -6), (3; -3,5), (4; 0), (5; 4,5), (6; 10). Эти точки лежат на кривой, которая является параболой, заданной уравнением $y = 0,5x^2 - 8$.</p><p>Ответ: точки (1; -7,5), (2; -6), (3; -3,5), (4; 0), (5; 4,5), (6; 10) лежат на параболе $y = 0,5x^2 - 8$.</p>" ] #original: array:6 [ "id" => 1421615 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034} "task" => array:2 [ "refs" => "199133" "type" => "task" ] "text" => "<p><strong>1)</strong> Последовательность Фибоначчи задается рекуррентной формулой $a_1 = 1$, $a_2 = 1$, $a_n = a_{n-1} + a_{n-2}$ для $n > 2$. Первые шесть членов последовательности: 1, 1, 2, 3, 5, 8. То есть, $a_1=1, a_2=1, a_3=2, a_4=3, a_5=5, a_6=8$.</p><p>Чтобы найти седьмой член последовательности, нужно сложить пятый и шестой члены:</p><p>$a_7 = a_5 + a_6 = 5 + 8 = 13$.</p><p>Координаты точки, изображающей седьмой член, равны номеру члена (абсцисса) и значению члена (ордината), то есть (7; 13).</p><p>Чтобы найти восьмой член последовательности, нужно сложить шестой и седьмой члены:</p><p>$a_8 = a_6 + a_7 = 8 + 13 = 21$.</p><p>Координаты точки, изображающей восьмой член, равны (8; 21).</p><p>Ответ: координаты седьмой точки (7; 13), восьмой точки (8; 21).</p><p><strong>2)</strong></p><p><strong>а)</strong> Для последовательности $a_n = n - 4$ найдем первые шесть членов:</p><p>$a_1 = 1 - 4 = -3$</p><p>$a_2 = 2 - 4 = -2$</p><p>$a_3 = 3 - 4 = -1$</p><p>$a_4 = 4 - 4 = 0$</p><p>$a_5 = 5 - 4 = 1$</p><p>$a_6 = 6 - 4 = 2$</p><p>Координаты соответствующих точек: (1; -3), (2; -2), (3; -1), (4; 0), (5; 1), (6; 2). Эти точки лежат на прямой, заданной уравнением $y = x - 4$.</p><p>Ответ: точки (1; -3), (2; -2), (3; -1), (4; 0), (5; 1), (6; 2) лежат на прямой $y = x - 4$.</p><p><strong>б)</strong> Для последовательности $a_n = \frac{n}{2}$ найдем первые шесть членов:</p><p>$a_1 = \frac{1}{2} = 0,5$</p><p>$a_2 = \frac{2}{2} = 1$</p><p>$a_3 = \frac{3}{2} = 1,5$</p><p>$a_4 = \frac{4}{2} = 2$</p><p>$a_5 = \frac{5}{2} = 2,5$</p><p>$a_6 = \frac{6}{2} = 3$</p><p>Координаты соответствующих точек: (1; 0,5), (2; 1), (3; 1,5), (4; 2), (5; 2,5), (6; 3). Эти точки лежат на прямой, заданной уравнением $y = \frac{x}{2}$ или $y = 0,5x$.</p><p>Ответ: точки (1; 0,5), (2; 1), (3; 1,5), (4; 2), (5; 2,5), (6; 3) лежат на прямой $y = \frac{x}{2}$.</p><p><strong>в)</strong> Для последовательности $a_n = \frac{12}{n}$ найдем первые шесть членов:</p><p>$a_1 = \frac{12}{1} = 12$</p><p>$a_2 = \frac{12}{2} = 6$</p><p>$a_3 = \frac{12}{3} = 4$</p><p>$a_4 = \frac{12}{4} = 3$</p><p>$a_5 = \frac{12}{5} = 2,4$</p><p>$a_6 = \frac{12}{6} = 2$</p><p>Координаты соответствующих точек: (1; 12), (2; 6), (3; 4), (4; 3), (5; 2,4), (6; 2). Эти точки лежат на кривой, которая является ветвью гиперболы, заданной уравнением $y = \frac{12}{x}$.</p><p>Ответ: точки (1; 12), (2; 6), (3; 4), (4; 3), (5; 2,4), (6; 2) лежат на гиперболе $y = \frac{12}{x}$.</p><p><strong>г)</strong> Для последовательности $a_n = 0,5n^2 - 8$ найдем первые шесть членов:</p><p>$a_1 = 0,5 \cdot 1^2 - 8 = 0,5 - 8 = -7,5$</p><p>$a_2 = 0,5 \cdot 2^2 - 8 = 0,5 \cdot 4 - 8 = 2 - 8 = -6$</p><p>$a_3 = 0,5 \cdot 3^2 - 8 = 0,5 \cdot 9 - 8 = 4,5 - 8 = -3,5$</p><p>$a_4 = 0,5 \cdot 4^2 - 8 = 0,5 \cdot 16 - 8 = 8 - 8 = 0$</p><p>$a_5 = 0,5 \cdot 5^2 - 8 = 0,5 \cdot 25 - 8 = 12,5 - 8 = 4,5$</p><p>$a_6 = 0,5 \cdot 6^2 - 8 = 0,5 \cdot 36 - 8 = 18 - 8 = 10$</p><p>Координаты соответствующих точек: (1; -7,5), (2; -6), (3; -3,5), (4; 0), (5; 4,5), (6; 10). Эти точки лежат на кривой, которая является параболой, заданной уравнением $y = 0,5x^2 - 8$.</p><p>Ответ: точки (1; -7,5), (2; -6), (3; -3,5), (4; 0), (5; 4,5), (6; 10) лежат на параболе $y = 0,5x^2 - 8$.</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" => "199134" "type" => "task" ] "previous" => array:2 [ "refs" => "199132" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\BookPage {#1042 #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" => 926479 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "133" "field_url" => "/9-klass/algebra/bunimovich-uchebnik/page-133" "field_display_title" => "133" "field_folder" => "1" "field_image_name" => "133" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Book {#1049} ] #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 {#1041 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Element {#1131 #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" => 1100677 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "926479" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1100677 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1133} "book_page" => array:2 [ "refs" => "926479" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "926480" "type" => "book_page" ] "previous" => array:2 [ "refs" => "926478" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:6 [ 0 => App\Models\Task {#1218 #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" => 199130 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "133" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-354" "field_display_title" => "354" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1220 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "199131" "type" => "task" ] "previous" => array:2 [ "refs" => "199129" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "926479" "type" => "book_page" ] ] #original: array:24 [ "id" => 199130 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "133" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-354" "field_display_title" => "354" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219} "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 {#1220} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1221} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1222} "content" => Illuminate\Database\Eloquent\Collection {#1225} "next" => array:2 [ "refs" => "199131" "type" => "task" ] "previous" => array:2 [ "refs" => "199129" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1231} "page" => array:2 [ "refs" => "926479" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1224 #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" => 199131 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "133" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-355" "field_display_title" => "355" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1230 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "199132" "type" => "task" ] "previous" => array:2 [ "refs" => "199130" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1245 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "926479" "type" => "book_page" ] ] #original: array:24 [ "id" => 199131 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "133" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-355" "field_display_title" => "355" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1223} "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 {#1230} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1232} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1228} "content" => Illuminate\Database\Eloquent\Collection {#1227} "next" => array:2 [ "refs" => "199132" "type" => "task" ] "previous" => array:2 [ "refs" => "199130" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1245} "page" => array:2 [ "refs" => "926479" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1229 #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" => 199132 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "133" "field_page_end" => null "field_url" => "/9-klass/algebra/bunimovich-uchebnik/01-356" "field_display_title" => "356" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1244 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1246 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "199133" "type" => "task" ] "previous" => array:2 [ "refs" => "199131" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1259 #items: array:1 [ …1] 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№357 (с. 133)
Условие. №357 (с. 133)
скриншот условия
357 1) Члены последовательности можно изображать точками на координатной плоскости: каждому члену соответствует точка, абсцисса которой равна его номеру, а ордината — значению этого члена. На рисунке 4.1 на координатной плоскости изображены первые шесть членов последовательности Фибоначчи: 1; 1; 2; 3; 5; 8. Им соответствуют точки: (1; 1), (2; 1), (3; 2), (4; 3), (5; 5), (6; 8). Назовите координаты точек, изображающих седьмой и восьмой члены этой последовательности.
2) Изобразите точками на координатной плоскости первые шесть членов последовательности $(a_n)$, если:
а) $a_n = n - 4$;
в) $a_n = \frac{12}{n}$;
б) $a_n = \frac{n}{2}$;
г) $a_n = 0,5n^2 - 8$.
В каждом случае скажите, на какой линии располагаются построенные точки; проведите тонко эту линию.
Решение. №357 (с. 133)
Решение 2. №357 (с. 133)
1) Последовательность Фибоначчи задается рекуррентной формулой $a_1 = 1$, $a_2 = 1$, $a_n = a_{n-1} + a_{n-2}$ для $n > 2$. Первые шесть членов последовательности: 1, 1, 2, 3, 5, 8. То есть, $a_1=1, a_2=1, a_3=2, a_4=3, a_5=5, a_6=8$.
Чтобы найти седьмой член последовательности, нужно сложить пятый и шестой члены:
$a_7 = a_5 + a_6 = 5 + 8 = 13$.
Координаты точки, изображающей седьмой член, равны номеру члена (абсцисса) и значению члена (ордината), то есть (7; 13).
Чтобы найти восьмой член последовательности, нужно сложить шестой и седьмой члены:
$a_8 = a_6 + a_7 = 8 + 13 = 21$.
Координаты точки, изображающей восьмой член, равны (8; 21).
Ответ: координаты седьмой точки (7; 13), восьмой точки (8; 21).
2)
а) Для последовательности $a_n = n - 4$ найдем первые шесть членов:
$a_1 = 1 - 4 = -3$
$a_2 = 2 - 4 = -2$
$a_3 = 3 - 4 = -1$
$a_4 = 4 - 4 = 0$
$a_5 = 5 - 4 = 1$
$a_6 = 6 - 4 = 2$
Координаты соответствующих точек: (1; -3), (2; -2), (3; -1), (4; 0), (5; 1), (6; 2). Эти точки лежат на прямой, заданной уравнением $y = x - 4$.
Ответ: точки (1; -3), (2; -2), (3; -1), (4; 0), (5; 1), (6; 2) лежат на прямой $y = x - 4$.
б) Для последовательности $a_n = \frac{n}{2}$ найдем первые шесть членов:
$a_1 = \frac{1}{2} = 0,5$
$a_2 = \frac{2}{2} = 1$
$a_3 = \frac{3}{2} = 1,5$
$a_4 = \frac{4}{2} = 2$
$a_5 = \frac{5}{2} = 2,5$
$a_6 = \frac{6}{2} = 3$
Координаты соответствующих точек: (1; 0,5), (2; 1), (3; 1,5), (4; 2), (5; 2,5), (6; 3). Эти точки лежат на прямой, заданной уравнением $y = \frac{x}{2}$ или $y = 0,5x$.
Ответ: точки (1; 0,5), (2; 1), (3; 1,5), (4; 2), (5; 2,5), (6; 3) лежат на прямой $y = \frac{x}{2}$.
в) Для последовательности $a_n = \frac{12}{n}$ найдем первые шесть членов:
$a_1 = \frac{12}{1} = 12$
$a_2 = \frac{12}{2} = 6$
$a_3 = \frac{12}{3} = 4$
$a_4 = \frac{12}{4} = 3$
$a_5 = \frac{12}{5} = 2,4$
$a_6 = \frac{12}{6} = 2$
Координаты соответствующих точек: (1; 12), (2; 6), (3; 4), (4; 3), (5; 2,4), (6; 2). Эти точки лежат на кривой, которая является ветвью гиперболы, заданной уравнением $y = \frac{12}{x}$.
Ответ: точки (1; 12), (2; 6), (3; 4), (4; 3), (5; 2,4), (6; 2) лежат на гиперболе $y = \frac{12}{x}$.
г) Для последовательности $a_n = 0,5n^2 - 8$ найдем первые шесть членов:
$a_1 = 0,5 \cdot 1^2 - 8 = 0,5 - 8 = -7,5$
$a_2 = 0,5 \cdot 2^2 - 8 = 0,5 \cdot 4 - 8 = 2 - 8 = -6$
$a_3 = 0,5 \cdot 3^2 - 8 = 0,5 \cdot 9 - 8 = 4,5 - 8 = -3,5$
$a_4 = 0,5 \cdot 4^2 - 8 = 0,5 \cdot 16 - 8 = 8 - 8 = 0$
$a_5 = 0,5 \cdot 5^2 - 8 = 0,5 \cdot 25 - 8 = 12,5 - 8 = 4,5$
$a_6 = 0,5 \cdot 6^2 - 8 = 0,5 \cdot 36 - 8 = 18 - 8 = 10$
Координаты соответствующих точек: (1; -7,5), (2; -6), (3; -3,5), (4; 0), (5; 4,5), (6; 10). Эти точки лежат на кривой, которая является параболой, заданной уравнением $y = 0,5x^2 - 8$.
Ответ: точки (1; -7,5), (2; -6), (3; -3,5), (4; 0), (5; 4,5), (6; 10) лежат на параболе $y = 0,5x^2 - 8$.
Другие задания:
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