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Номер 620, страница 177 - гдз по алгебре 9 класс учебник Макарычев, Миндюк
Авторы: Макарычев Ю. Н., Миндюк Н. Г., Нешков К. И., Суворова С. Б.
Тип: Учебник
Издательство: Просвещение
Год издания: 2023 - 2026
Уровень обучения: базовый
Цвет обложки: белый, зелёный, фиолетовый
ISBN: 978-5-09-112135-3
Допущено Министерством просвещения Российской Федерации
Популярные ГДЗ в 9 классе
Глава 5. Арифметическая и геометрическая прогрессии. Параграф 10. Геометрическая прогрессия. 30. Формула суммы первых n членов геометрической прогрессии - номер 620, страница 177.
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" => 171806 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "177" "field_page_end" => null "field_url" => "/9-klass/algebra/makarychev-fgos/620" "field_display_title" => "620" "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 {#1080 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 171142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Арифметическая и геометрическая прогрессии" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #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 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_page_start" => "149" "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 {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 171142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Арифметическая и геометрическая прогрессии" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "149" "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 {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1077} ] #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 {#1203 #items: array:3 [ 0 => App\Models\Branch {#1213 #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" => 171142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Арифметическая и геометрическая прогрессии" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Term {#1214 #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 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_page_start" => "149" "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 {#1254 #items: array:1 [ 0 => App\Models\Book {#1216 #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" => 10 "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 {#1218 #items: array:1 [ 0 => App\Models\Term {#1215 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Term {#1220 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ 0 => App\Models\Term {#1222 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:4 [ 0 => App\Models\Term {#1224 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1225 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1226 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1227 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1228 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Term {#1230 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ 0 => App\Models\Term {#1232 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Term {#1234 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1235 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1236 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Term {#1238 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Term {#1240 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "16" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ 0 => App\Models\Term {#1242 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1243 #items: array:1 [ 0 => App\Models\Term {#1244 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2023" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => null "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "934" "field_priority" => "2" "field_default_folder" => "/algebra_09/makarychev-2023/" "field_isbn" => "978-5-09-112135-3" "field_cover" => array:1 [ 0 => "/media/algebra_09/makarychev-2023/covers/cover1.webp?ts=1734038732" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/makarychev-2023/covers/cover1.webp?ts=1734038732" "alt" => "" "width" => "680" "height" => "1065" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1245 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1246 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1247 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/makarychev-fgos" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:3 [ 0 => App\Models\Term {#1249 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1250 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1251 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => "<!--Текст в низу страницы--> <h2> Учебный процесс с решебником Ключниковой </h2> <p> Изучение алгебры всегда связано с различными недопониманиями. Довольно часто у ребят не хватает времени, чтобы детально проработать каждый параграф. Теорию они проходят в спешке, так как основное внимание сосредоточено в основном на практических заданиях, которые учителя задают на дом в больших количествах. ГДЗ «<strong>Алгебра Учебник Макарычев, Миндюк, Нешков 9 класс (Просвещение, 2023г.)</strong>» позволит школьникам сбалансировать учебный процесс, не упуская ни одной мелочи. </p> <p> Многие ученики испытывают проблемы с пониманием таких аспектов: </p> <ol> <li>Действия над действительными числами.</li> <li>Погрешность и точность приближения?</li> <li>Свойства чётности и нечётности функций.</li> <li>Целое уравнение и его корни.</li> <li>Решение неравенств методом интервалов.</li> <li>Уравнение с двумя переменными и его график.</li> </ol> <p> Современные учебники часто снабжены недостаточно развернутой теорией, что мешает учащимся еще раз пробежаться по материалу, если они пропустили урок или не поняли объяснения преподавателя. Поэтому иногда подростки приступают к решению задач так до конца и не разобравшись в параграфах. Неудивительно, что потом в их работах обнаруживается много ошибок. Благодаря решебнику можно избежать любых недочетов при выполнении номеров. </p> <h2> Почему работать с решебником нужно правильно? </h2> <p> Школьники всегда делятся на две категории. Первые, даже с большим скрипом, но находят решения и ответы самостоятельно, а вот вторые постоянно списывают домашние задания и ничего толком в дисциплине не понимают. Но используя <strong>решебник по алгебре за 9 класс к учебнику Макарычева Ю.Н., Миндюка Н.Г., Нешкова К.И. (Просвещение)</strong> все ребята получают возможность полноценно разобраться в программе этого курса. </p> <p> Неправильное взаимодействие с пособием может привести к следующим последствиям: </p> <ol> <li>материал не будет освоен полностью;</li> <li>постоянные ошибки приведут к снижению успеваемости;</li> <li>отсутствие понимания даже элементарных уравнений;</li> <li>проваленные экзамены.</li> </ol> <p> Соблазн просто списать нужное упражнение несомненно очень велик. Но не стоит забывать о том, что тогда не удастся полностью понять суть изучаемой темы, а это непременно аукнется довольно быстро плохими оценками. Поэтому всегда стоит применять решебник строго по назначению - для самопроверки и работы над ошибками. </p> <h2> Хорошая подготовка д/з с ГДЗ к учебнику Макарычева </h2> <p> Любой учебный процесс - дело непредсказуемое. Даже отличники не застрахованы от получения двоек. Поэтому школьникам не только стоит быть предельно внимательными, но и систематически проверять себя по <strong>готовым решениям к учебнику Макарычева за 9 класс</strong>, написанными в соответствии с ФГОС. Сделать это очень просто, так как пособие находится в онлайн-доступе, и открыть его можно с любого гаджета, подключенного к Интернету. Тренировки со справочником не отнимают много времени, так как авторы снабдили его подробными решениями, верными ответами и многочисленными наглядными примерами, которые легко понять и запомнить. </p> <p> Онлайн-издание пригодится учащимся для многого: </p> <ul> <li>сверки д/з;</li> <li>подготовки к тестированиям;</li> <li>разбора проблематичных параграфов;</li> <li>проработки различных упущений, и т.д.</li> </ul> <p> Некоторые учителя сами советуют своим подопечным использовать решебник, так как прекрасно понимают, что иначе дети просто не справятся с программой. Конечно, есть и еще один вариант - занятия с репетитором. Но в наше время сложно найти преподавателя, который бы смог не только найти общий язык с учеником, но и доступно излагал материал. Кроме того, дополнительные уроки тоже отнимают уйму времени, а именно его у подростков и так мало. 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Довольно часто у ребят не хватает времени, чтобы детально проработать каждый параграф. Теорию они проходят в спешке, так как основное внимание сосредоточено в основном на практических заданиях, которые учителя задают на дом в больших количествах. ГДЗ «<strong>Алгебра Учебник Макарычев, Миндюк, Нешков 9 класс (Просвещение, 2023г.)</strong>» позволит школьникам сбалансировать учебный процесс, не упуская ни одной мелочи. </p> <p> Многие ученики испытывают проблемы с пониманием таких аспектов: </p> <ol> <li>Действия над действительными числами.</li> <li>Погрешность и точность приближения?</li> <li>Свойства чётности и нечётности функций.</li> <li>Целое уравнение и его корни.</li> <li>Решение неравенств методом интервалов.</li> <li>Уравнение с двумя переменными и его график.</li> </ol> <p> Современные учебники часто снабжены недостаточно развернутой теорией, что мешает учащимся еще раз пробежаться по материалу, если они пропустили урок или не поняли объяснения преподавателя. 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Поэтому школьникам не только стоит быть предельно внимательными, но и систематически проверять себя по <strong>готовым решениям к учебнику Макарычева за 9 класс</strong>, написанными в соответствии с ФГОС. Сделать это очень просто, так как пособие находится в онлайн-доступе, и открыть его можно с любого гаджета, подключенного к Интернету. Тренировки со справочником не отнимают много времени, так как авторы снабдили его подробными решениями, верными ответами и многочисленными наглядными примерами, которые легко понять и запомнить. </p> <p> Онлайн-издание пригодится учащимся для многого: </p> <ul> <li>сверки д/з;</li> <li>подготовки к тестированиям;</li> <li>разбора проблематичных параграфов;</li> <li>проработки различных упущений, и т.д.</li> </ul> <p> Некоторые учителя сами советуют своим подопечным использовать решебник, так как прекрасно понимают, что иначе дети просто не справятся с программой. Конечно, есть и еще один вариант - занятия с репетитором. 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"branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 171150 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. Формула суммы первых n членов геометрической прогрессии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1260 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "174" "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 {#1261 #items: array:1 [ 0 => App\Models\Book {#1216} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1346 #items: array:1 [ 0 => App\Models\Branch {#1262 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 171150 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. Формула суммы первых n членов геометрической прогрессии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1260} "field_page_start" => "174" "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 {#1261} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1346} ] #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 {#1191 #items: array:9 [ 0 => App\Models\Element {#1175 #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" => 179067 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Edition {#1174 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "text" => """ <p><strong>620</strong>. Найдите сумму первых семи членов геометрической прогрессии <em>(bₙ),</em> если:</p><figure><figure>\n <img src="/media/algebra_09/makarychev-2023/0-00/620.webp?ts=1744048453" alt="Найти сумму первых семи членов геометрической прогрессии" loading="lazy" width="1040" height="267">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "620-1.jpg" "alt" => null "width" => "1636" "height" => 368 "path" => "/media/algebra_09/makarychev-2023/0-00/620-1.webp?ts=1734090832" ] ] ] #original: array:7 [ "id" => 179067 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "text" => """ <p><strong>620</strong>. Найдите сумму первых семи членов геометрической прогрессии <em>(bₙ),</em> если:</p><figure><figure>\n <img src="/media/algebra_09/makarychev-2023/0-00/620.webp?ts=1744048453" alt="Найти сумму первых семи членов геометрической прогрессии" loading="lazy" width="1040" height="267">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "620-1.jpg" "alt" => null "width" => "1636" "height" => 368 "path" => "/media/algebra_09/makarychev-2023/0-00/620-1.webp?ts=1734090832" ] ] ] #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 {#1166 #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" => 180056 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Edition {#1167 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "620-1.jpg" "alt" => null "width" => "1624" "height" => 1009 "path" => "/media/algebra_09/makarychev-2023/1-00/620-1.webp?ts=1734092032" ] 1 => array:5 [ "name" => "620-2.jpg" "alt" => null "width" => "1624" "height" => 2282 "path" => "/media/algebra_09/makarychev-2023/1-00/620-2.webp?ts=1734092032" ] 2 => array:5 [ "name" => "620-3.jpg" "alt" => null "width" => "1624" "height" => 1609 "path" => "/media/algebra_09/makarychev-2023/1-00/620-3.webp?ts=1734092032" ] ] ] #original: array:6 [ "id" => 180056 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "620-1.jpg" "alt" => null "width" => "1624" "height" => 1009 "path" => "/media/algebra_09/makarychev-2023/1-00/620-1.webp?ts=1734092032" ] 1 => array:5 [ "name" => "620-2.jpg" "alt" => null "width" => "1624" "height" => 2282 "path" => "/media/algebra_09/makarychev-2023/1-00/620-2.webp?ts=1734092032" ] 2 => array:5 [ "name" => "620-3.jpg" "alt" => null "width" => "1624" "height" => 1609 "path" => "/media/algebra_09/makarychev-2023/1-00/620-3.webp?ts=1734092032" ] ] ] #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 {#1158 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 180789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Edition {#1157 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:4 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "700" "height" => 1562 "path" => "/media/algebra_09/makarychev-2023/2-00/620-1.webp?ts=1734091838" ] 1 => array:5 [ "name" => "620-2.png" "alt" => null "width" => "700" "height" => 1601 "path" => "/media/algebra_09/makarychev-2023/2-00/620-2.webp?ts=1734091838" ] 2 => array:5 [ "name" => "620-3.png" "alt" => null "width" => "700" "height" => 1593 "path" => "/media/algebra_09/makarychev-2023/2-00/620-3.webp?ts=1734091838" ] 3 => array:5 [ "name" => "620-4.png" "alt" => null "width" => "700" "height" => 1602 "path" => "/media/algebra_09/makarychev-2023/2-00/620-4.webp?ts=1734091838" ] ] ] #original: array:6 [ "id" => 180789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1148} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:4 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "700" "height" => 1562 "path" => "/media/algebra_09/makarychev-2023/2-00/620-1.webp?ts=1734091838" ] 1 => array:5 [ "name" => "620-2.png" "alt" => null "width" => "700" "height" => 1601 "path" => "/media/algebra_09/makarychev-2023/2-00/620-2.webp?ts=1734091838" ] 2 => array:5 [ "name" => "620-3.png" "alt" => null "width" => "700" "height" => 1593 "path" => "/media/algebra_09/makarychev-2023/2-00/620-3.webp?ts=1734091838" ] 3 => array:5 [ "name" => "620-4.png" "alt" => null "width" => "700" "height" => 1602 "path" => "/media/algebra_09/makarychev-2023/2-00/620-4.webp?ts=1734091838" ] ] ] #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\Element {#1150 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 181471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Edition {#1147 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "650" "height" => 917 "path" => "/media/algebra_09/makarychev-2023/3-00/620-1.webp?ts=1734091800" ] ] ] #original: array:6 [ "id" => 181471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1140} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "650" "height" => 917 "path" => "/media/algebra_09/makarychev-2023/3-00/620-1.webp?ts=1734091800" ] ] ] #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\Element {#1142 #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" => 182142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Edition {#1139 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "595" "height" => 1960 "path" => "/media/algebra_09/makarychev-2023/4-00/620-1.webp?ts=1734091935" ] ] ] #original: array:6 [ "id" => 182142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1132} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "595" "height" => 1960 "path" => "/media/algebra_09/makarychev-2023/4-00/620-1.webp?ts=1734091935" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1134 #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" => 182904 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1133 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "586" "height" => 369 "path" => "/media/algebra_09/makarychev-2023/5-00/620-1.webp?ts=1734092827" ] ] ] #original: array:6 [ "id" => 182904 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "586" "height" => 369 "path" => "/media/algebra_09/makarychev-2023/5-00/620-1.webp?ts=1734092827" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Element {#1126 #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" => 183579 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1121 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "1345" "height" => 507 "path" => "/media/algebra_09/makarychev-2023/6-00/620-1.webp?ts=1734092996" ] ] ] #original: array:6 [ "id" => 183579 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.png" "alt" => null "width" => "1345" "height" => 507 "path" => "/media/algebra_09/makarychev-2023/6-00/620-1.webp?ts=1734092996" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 7 => App\Models\Element {#1118 #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" => 184058 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1106 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.jpg" "alt" => null "width" => "1575" "height" => 449 "path" => "/media/algebra_09/makarychev-2023/7-00/620-1.webp?ts=1734093415" ] ] ] #original: array:6 [ "id" => 184058 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1106} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "620-1.jpg" "alt" => null "width" => "1575" "height" => 449 "path" => "/media/algebra_09/makarychev-2023/7-00/620-1.webp?ts=1734093415" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 8 => App\Models\Element {#1108 #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" => 1349366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "171806" "type" => "task" ] "text" => "<p>Для нахождения суммы первых семи членов геометрической прогрессии $S_7$ используется формула $S_n = \frac{b_1(q^n - 1)}{q - 1}$ или $S_n = \frac{b_1(1 - q^n)}{1 - q}$, где $b_1$ — первый член прогрессии, $q$ — знаменатель прогрессии, $n$ — количество членов. Во всех случаях $n=7$.<br>Сначала необходимо найти первый член прогрессии $b_1$, используя формулу n-го члена $b_n = b_1 \cdot q^{n-1}$.</p><p><strong>а)</strong> $b_7 = 72.9$, $q = 1.5$</p><p>1. Найдем первый член прогрессии $b_1$.<br>Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=7$ имеем: $b_7 = b_1 \cdot q^{6}$.<br>Подставим заданные значения: $72.9 = b_1 \cdot (1.5)^6$.<br>Вычислим $(1.5)^6$: $1.5 = \frac{3}{2}$, поэтому $(1.5)^6 = (\frac{3}{2})^6 = \frac{3^6}{2^6} = \frac{729}{64}$.<br>Теперь решим уравнение для $b_1$: $\frac{729}{10} = b_1 \cdot \frac{729}{64}$.<br>Отсюда $b_1 = \frac{729}{10} \div \frac{729}{64} = \frac{729}{10} \cdot \frac{64}{729} = \frac{64}{10} = 6.4$.</p><p>2. Найдем сумму первых семи членов $S_7$.<br>Используем формулу $S_7 = \frac{b_1(q^7 - 1)}{q - 1}$.<br>Вычислим $q^7$: $(1.5)^7 = (\frac{3}{2})^7 = \frac{3^7}{2^7} = \frac{2187}{128}$.<br>$S_7 = \frac{6.4((1.5)^7 - 1)}{1.5 - 1} = \frac{6.4(\frac{2187}{128} - 1)}{0.5} = \frac{\frac{64}{10}(\frac{2187 - 128}{128})}{\frac{1}{2}} = \frac{\frac{64}{10} \cdot \frac{2059}{128}}{\frac{1}{2}}$.<br>$S_7 = \frac{64 \cdot 2059}{10 \cdot 128} \cdot 2 = \frac{1 \cdot 2059}{10 \cdot 2} \cdot 2 = \frac{2059}{10} = 205.9$.<br>Ответ: $205.9$</p><p><strong>б)</strong> $b_5 = \frac{16}{9}$, $q = \frac{2}{3}$</p><p>1. Найдем первый член прогрессии $b_1$.<br>Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=5$ имеем: $b_5 = b_1 \cdot q^{4}$.<br>Подставим заданные значения: $\frac{16}{9} = b_1 \cdot (\frac{2}{3})^4$.<br>Вычислим $q^4$: $(\frac{2}{3})^4 = \frac{2^4}{3^4} = \frac{16}{81}$.<br>Решим уравнение: $\frac{16}{9} = b_1 \cdot \frac{16}{81}$.<br>Отсюда $b_1 = \frac{16}{9} \div \frac{16}{81} = \frac{16}{9} \cdot \frac{81}{16} = \frac{81}{9} = 9$.</p><p>2. Найдем сумму первых семи членов $S_7$.<br>Поскольку $|q| < 1$, удобнее использовать формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.<br>Вычислим $q^7$: $(\frac{2}{3})^7 = \frac{2^7}{3^7} = \frac{128}{2187}$.<br>$S_7 = \frac{9(1 - (\frac{2}{3})^7)}{1 - \frac{2}{3}} = \frac{9(1 - \frac{128}{2187})}{\frac{1}{3}} = \frac{9(\frac{2187 - 128}{2187})}{\frac{1}{3}} = \frac{9 \cdot \frac{2059}{2187}}{\frac{1}{3}}$.<br>$S_7 = 9 \cdot \frac{2059}{2187} \cdot 3 = 27 \cdot \frac{2059}{2187}$.<br>Так как $2187 = 3^7 = 27 \cdot 81$, то $S_7 = 27 \cdot \frac{2059}{27 \cdot 81} = \frac{2059}{81}$.<br>Ответ: $\frac{2059}{81}$</p><p><strong>в)</strong> $b_3 = 64$, $q = \frac{1}{2}$</p><p>1. Найдем первый член прогрессии $b_1$.<br>Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=3$ имеем: $b_3 = b_1 \cdot q^{2}$.<br>Подставим заданные значения: $64 = b_1 \cdot (\frac{1}{2})^2 = b_1 \cdot \frac{1}{4}$.<br>Отсюда $b_1 = 64 \cdot 4 = 256$.</p><p>2. Найдем сумму первых семи членов $S_7$.<br>Используем формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.<br>Вычислим $q^7$: $(\frac{1}{2})^7 = \frac{1}{128}$.<br>$S_7 = \frac{256(1 - (\frac{1}{2})^7)}{1 - \frac{1}{2}} = \frac{256(1 - \frac{1}{128})}{\frac{1}{2}} = \frac{256(\frac{127}{128})}{\frac{1}{2}}$.<br>$S_7 = \frac{\frac{256 \cdot 127}{128}}{\frac{1}{2}} = \frac{2 \cdot 127}{\frac{1}{2}} = 2 \cdot 127 \cdot 2 = 508$.<br>Ответ: $508$</p><p><strong>г)</strong> $b_4 = 81$, $q = -\frac{1}{3}$</p><p>1. Найдем первый член прогрессии $b_1$.<br>Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=4$ имеем: $b_4 = b_1 \cdot q^{3}$.<br>Подставим заданные значения: $81 = b_1 \cdot (-\frac{1}{3})^3 = b_1 \cdot (-\frac{1}{27})$.<br>Отсюда $b_1 = 81 \cdot (-27) = -2187$.</p><p>2. Найдем сумму первых семи членов $S_7$.<br>Используем формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.<br>Вычислим $q^7$: $(-\frac{1}{3})^7 = -\frac{1}{3^7} = -\frac{1}{2187}$.<br>$S_7 = \frac{-2187(1 - (-\frac{1}{2187}))}{1 - (-\frac{1}{3})} = \frac{-2187(1 + \frac{1}{2187})}{1 + \frac{1}{3}} = \frac{-2187(\frac{2187+1}{2187})}{\frac{4}{3}}$.<br>$S_7 = \frac{-2187 \cdot \frac{2188}{2187}}{\frac{4}{3}} = \frac{-2188}{\frac{4}{3}} = -2188 \cdot \frac{3}{4}$.<br>Сократив 2188 на 4, получим 547.<br>$S_7 = -547 \cdot 3 = -1641$.<br>Ответ: $-1641$</p>" ] #original: array:6 [ "id" => 1349366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "171806" "type" => "task" ] "text" => "<p>Для нахождения суммы первых семи членов геометрической прогрессии $S_7$ используется формула $S_n = \frac{b_1(q^n - 1)}{q - 1}$ или $S_n = \frac{b_1(1 - q^n)}{1 - q}$, где $b_1$ — первый член прогрессии, $q$ — знаменатель прогрессии, $n$ — количество членов. Во всех случаях $n=7$.<br>Сначала необходимо найти первый член прогрессии $b_1$, используя формулу n-го члена $b_n = b_1 \cdot q^{n-1}$.</p><p><strong>а)</strong> $b_7 = 72.9$, $q = 1.5$</p><p>1. Найдем первый член прогрессии $b_1$.<br>Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=7$ имеем: $b_7 = b_1 \cdot q^{6}$.<br>Подставим заданные значения: $72.9 = b_1 \cdot (1.5)^6$.<br>Вычислим $(1.5)^6$: $1.5 = \frac{3}{2}$, поэтому $(1.5)^6 = (\frac{3}{2})^6 = \frac{3^6}{2^6} = \frac{729}{64}$.<br>Теперь решим уравнение для $b_1$: $\frac{729}{10} = b_1 \cdot \frac{729}{64}$.<br>Отсюда $b_1 = \frac{729}{10} \div \frac{729}{64} = \frac{729}{10} \cdot \frac{64}{729} = \frac{64}{10} = 6.4$.</p><p>2. Найдем сумму первых семи членов $S_7$.<br>Используем формулу $S_7 = \frac{b_1(q^7 - 1)}{q - 1}$.<br>Вычислим $q^7$: $(1.5)^7 = (\frac{3}{2})^7 = \frac{3^7}{2^7} = \frac{2187}{128}$.<br>$S_7 = \frac{6.4((1.5)^7 - 1)}{1.5 - 1} = \frac{6.4(\frac{2187}{128} - 1)}{0.5} = \frac{\frac{64}{10}(\frac{2187 - 128}{128})}{\frac{1}{2}} = \frac{\frac{64}{10} \cdot \frac{2059}{128}}{\frac{1}{2}}$.<br>$S_7 = \frac{64 \cdot 2059}{10 \cdot 128} \cdot 2 = \frac{1 \cdot 2059}{10 \cdot 2} \cdot 2 = \frac{2059}{10} = 205.9$.<br>Ответ: $205.9$</p><p><strong>б)</strong> $b_5 = \frac{16}{9}$, $q = \frac{2}{3}$</p><p>1. Найдем первый член прогрессии $b_1$.<br>Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=5$ имеем: $b_5 = b_1 \cdot q^{4}$.<br>Подставим заданные значения: $\frac{16}{9} = b_1 \cdot (\frac{2}{3})^4$.<br>Вычислим $q^4$: $(\frac{2}{3})^4 = \frac{2^4}{3^4} = \frac{16}{81}$.<br>Решим уравнение: $\frac{16}{9} = b_1 \cdot \frac{16}{81}$.<br>Отсюда $b_1 = \frac{16}{9} \div \frac{16}{81} = \frac{16}{9} \cdot \frac{81}{16} = \frac{81}{9} = 9$.</p><p>2. Найдем сумму первых семи членов $S_7$.<br>Поскольку $|q| < 1$, удобнее использовать формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.<br>Вычислим $q^7$: $(\frac{2}{3})^7 = \frac{2^7}{3^7} = \frac{128}{2187}$.<br>$S_7 = \frac{9(1 - (\frac{2}{3})^7)}{1 - \frac{2}{3}} = \frac{9(1 - \frac{128}{2187})}{\frac{1}{3}} = \frac{9(\frac{2187 - 128}{2187})}{\frac{1}{3}} = \frac{9 \cdot \frac{2059}{2187}}{\frac{1}{3}}$.<br>$S_7 = 9 \cdot \frac{2059}{2187} \cdot 3 = 27 \cdot \frac{2059}{2187}$.<br>Так как $2187 = 3^7 = 27 \cdot 81$, то $S_7 = 27 \cdot \frac{2059}{27 \cdot 81} = \frac{2059}{81}$.<br>Ответ: $\frac{2059}{81}$</p><p><strong>в)</strong> $b_3 = 64$, $q = \frac{1}{2}$</p><p>1. Найдем первый член прогрессии $b_1$.<br>Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=3$ имеем: $b_3 = b_1 \cdot q^{2}$.<br>Подставим заданные значения: $64 = b_1 \cdot (\frac{1}{2})^2 = b_1 \cdot \frac{1}{4}$.<br>Отсюда $b_1 = 64 \cdot 4 = 256$.</p><p>2. Найдем сумму первых семи членов $S_7$.<br>Используем формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.<br>Вычислим $q^7$: $(\frac{1}{2})^7 = \frac{1}{128}$.<br>$S_7 = \frac{256(1 - (\frac{1}{2})^7)}{1 - \frac{1}{2}} = \frac{256(1 - \frac{1}{128})}{\frac{1}{2}} = \frac{256(\frac{127}{128})}{\frac{1}{2}}$.<br>$S_7 = \frac{\frac{256 \cdot 127}{128}}{\frac{1}{2}} = \frac{2 \cdot 127}{\frac{1}{2}} = 2 \cdot 127 \cdot 2 = 508$.<br>Ответ: $508$</p><p><strong>г)</strong> $b_4 = 81$, $q = -\frac{1}{3}$</p><p>1. Найдем первый член прогрессии $b_1$.<br>Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=4$ имеем: $b_4 = b_1 \cdot q^{3}$.<br>Подставим заданные значения: $81 = b_1 \cdot (-\frac{1}{3})^3 = b_1 \cdot (-\frac{1}{27})$.<br>Отсюда $b_1 = 81 \cdot (-27) = -2187$.</p><p>2. Найдем сумму первых семи членов $S_7$.<br>Используем формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.<br>Вычислим $q^7$: $(-\frac{1}{3})^7 = -\frac{1}{3^7} = -\frac{1}{2187}$.<br>$S_7 = \frac{-2187(1 - (-\frac{1}{2187}))}{1 - (-\frac{1}{3})} = \frac{-2187(1 + \frac{1}{2187})}{1 + \frac{1}{3}} = \frac{-2187(\frac{2187+1}{2187})}{\frac{4}{3}}$.<br>$S_7 = \frac{-2187 \cdot \frac{2188}{2187}}{\frac{4}{3}} = \frac{-2188}{\frac{4}{3}} = -2188 \cdot \frac{3}{4}$.<br>Сократив 2188 на 4, получим 547.<br>$S_7 = -547 \cdot 3 = -1641$.<br>Ответ: $-1641$</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" => "171807" "type" => "task" ] "previous" => array:2 [ "refs" => "171805" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Book {#1216} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\BookPage {#1194 #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" => 1029915 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "177" "field_url" => "/9-klass/algebra/makarychev-fgos/page-177" "field_display_title" => "177" "field_folder" => "1" "field_image_name" => "177" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1195 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Book {#1216} ] #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 {#1160 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Element {#1177 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1029916" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1029914" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1557 #items: array:8 [ 0 => App\Models\Task {#1571 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#1584 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#1581 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1577 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1587 #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 [ …24] #original: array:24 [ …24] …16 } 5 => App\Models\Task {#1594 …30} 6 => App\Models\Task {#1655 …30} 7 => App\Models\Task {#1708 …30} ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1029915 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "177" "field_url" => "/9-klass/algebra/makarychev-fgos/page-177" "field_display_title" => "177" "field_folder" => "1" "field_image_name" => "177" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1195} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1210} "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 {#1160} "content" => Illuminate\Database\Eloquent\Collection {#1179} "next" => array:2 [ "refs" => "1029916" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1029914" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1557} ] #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" => 171806 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "177" "field_page_end" => null "field_url" => "/9-klass/algebra/makarychev-fgos/620" "field_display_title" => "620" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1080} "parent_branches" => 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№620 (с. 177)
Условие. №620 (с. 177)
скриншот условия
620. Найдите сумму первых семи членов геометрической прогрессии (bₙ), если:
Решение 1. №620 (с. 177)
Решение 2. №620 (с. 177)
Решение 3. №620 (с. 177)
Решение 4. №620 (с. 177)
Решение 5. №620 (с. 177)
Решение 6. №620 (с. 177)
Решение 7. №620 (с. 177)
Решение 8. №620 (с. 177)
Для нахождения суммы первых семи членов геометрической прогрессии $S_7$ используется формула $S_n = \frac{b_1(q^n - 1)}{q - 1}$ или $S_n = \frac{b_1(1 - q^n)}{1 - q}$, где $b_1$ — первый член прогрессии, $q$ — знаменатель прогрессии, $n$ — количество членов. Во всех случаях $n=7$.
Сначала необходимо найти первый член прогрессии $b_1$, используя формулу n-го члена $b_n = b_1 \cdot q^{n-1}$.
а) $b_7 = 72.9$, $q = 1.5$
1. Найдем первый член прогрессии $b_1$.
Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=7$ имеем: $b_7 = b_1 \cdot q^{6}$.
Подставим заданные значения: $72.9 = b_1 \cdot (1.5)^6$.
Вычислим $(1.5)^6$: $1.5 = \frac{3}{2}$, поэтому $(1.5)^6 = (\frac{3}{2})^6 = \frac{3^6}{2^6} = \frac{729}{64}$.
Теперь решим уравнение для $b_1$: $\frac{729}{10} = b_1 \cdot \frac{729}{64}$.
Отсюда $b_1 = \frac{729}{10} \div \frac{729}{64} = \frac{729}{10} \cdot \frac{64}{729} = \frac{64}{10} = 6.4$.
2. Найдем сумму первых семи членов $S_7$.
Используем формулу $S_7 = \frac{b_1(q^7 - 1)}{q - 1}$.
Вычислим $q^7$: $(1.5)^7 = (\frac{3}{2})^7 = \frac{3^7}{2^7} = \frac{2187}{128}$.
$S_7 = \frac{6.4((1.5)^7 - 1)}{1.5 - 1} = \frac{6.4(\frac{2187}{128} - 1)}{0.5} = \frac{\frac{64}{10}(\frac{2187 - 128}{128})}{\frac{1}{2}} = \frac{\frac{64}{10} \cdot \frac{2059}{128}}{\frac{1}{2}}$.
$S_7 = \frac{64 \cdot 2059}{10 \cdot 128} \cdot 2 = \frac{1 \cdot 2059}{10 \cdot 2} \cdot 2 = \frac{2059}{10} = 205.9$.
Ответ: $205.9$
б) $b_5 = \frac{16}{9}$, $q = \frac{2}{3}$
1. Найдем первый член прогрессии $b_1$.
Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=5$ имеем: $b_5 = b_1 \cdot q^{4}$.
Подставим заданные значения: $\frac{16}{9} = b_1 \cdot (\frac{2}{3})^4$.
Вычислим $q^4$: $(\frac{2}{3})^4 = \frac{2^4}{3^4} = \frac{16}{81}$.
Решим уравнение: $\frac{16}{9} = b_1 \cdot \frac{16}{81}$.
Отсюда $b_1 = \frac{16}{9} \div \frac{16}{81} = \frac{16}{9} \cdot \frac{81}{16} = \frac{81}{9} = 9$.
2. Найдем сумму первых семи членов $S_7$.
Поскольку $|q| < 1$, удобнее использовать формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.
Вычислим $q^7$: $(\frac{2}{3})^7 = \frac{2^7}{3^7} = \frac{128}{2187}$.
$S_7 = \frac{9(1 - (\frac{2}{3})^7)}{1 - \frac{2}{3}} = \frac{9(1 - \frac{128}{2187})}{\frac{1}{3}} = \frac{9(\frac{2187 - 128}{2187})}{\frac{1}{3}} = \frac{9 \cdot \frac{2059}{2187}}{\frac{1}{3}}$.
$S_7 = 9 \cdot \frac{2059}{2187} \cdot 3 = 27 \cdot \frac{2059}{2187}$.
Так как $2187 = 3^7 = 27 \cdot 81$, то $S_7 = 27 \cdot \frac{2059}{27 \cdot 81} = \frac{2059}{81}$.
Ответ: $\frac{2059}{81}$
в) $b_3 = 64$, $q = \frac{1}{2}$
1. Найдем первый член прогрессии $b_1$.
Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=3$ имеем: $b_3 = b_1 \cdot q^{2}$.
Подставим заданные значения: $64 = b_1 \cdot (\frac{1}{2})^2 = b_1 \cdot \frac{1}{4}$.
Отсюда $b_1 = 64 \cdot 4 = 256$.
2. Найдем сумму первых семи членов $S_7$.
Используем формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.
Вычислим $q^7$: $(\frac{1}{2})^7 = \frac{1}{128}$.
$S_7 = \frac{256(1 - (\frac{1}{2})^7)}{1 - \frac{1}{2}} = \frac{256(1 - \frac{1}{128})}{\frac{1}{2}} = \frac{256(\frac{127}{128})}{\frac{1}{2}}$.
$S_7 = \frac{\frac{256 \cdot 127}{128}}{\frac{1}{2}} = \frac{2 \cdot 127}{\frac{1}{2}} = 2 \cdot 127 \cdot 2 = 508$.
Ответ: $508$
г) $b_4 = 81$, $q = -\frac{1}{3}$
1. Найдем первый член прогрессии $b_1$.
Из формулы $b_n = b_1 \cdot q^{n-1}$ для $n=4$ имеем: $b_4 = b_1 \cdot q^{3}$.
Подставим заданные значения: $81 = b_1 \cdot (-\frac{1}{3})^3 = b_1 \cdot (-\frac{1}{27})$.
Отсюда $b_1 = 81 \cdot (-27) = -2187$.
2. Найдем сумму первых семи членов $S_7$.
Используем формулу $S_7 = \frac{b_1(1 - q^7)}{1 - q}$.
Вычислим $q^7$: $(-\frac{1}{3})^7 = -\frac{1}{3^7} = -\frac{1}{2187}$.
$S_7 = \frac{-2187(1 - (-\frac{1}{2187}))}{1 - (-\frac{1}{3})} = \frac{-2187(1 + \frac{1}{2187})}{1 + \frac{1}{3}} = \frac{-2187(\frac{2187+1}{2187})}{\frac{4}{3}}$.
$S_7 = \frac{-2187 \cdot \frac{2188}{2187}}{\frac{4}{3}} = \frac{-2188}{\frac{4}{3}} = -2188 \cdot \frac{3}{4}$.
Сократив 2188 на 4, получим 547.
$S_7 = -547 \cdot 3 = -1641$.
Ответ: $-1641$
Другие задания:
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