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Консультация № 198778
28.05.2020, 16:14
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Здравствуйте, уважаемые эксперты! Помогите, пожалуйста, решить задачу:
Последовательность a(n)=2a(n-1)+15a(n-2) и начальными условиями a(0)=-6; a(1)=2.
a)Найти производящую функцию последовательности.
b)С помощью найденной функции построить явную формулу для последовательности.
Последовательность a(n)=2a(n-1)+15a(n-2) и начальными условиями a(0)=-6; a(1)=2.
a)Найти производящую функцию последовательности.
b)С помощью найденной функции построить явную формулу для последовательности.
Обсуждение
02.06.2020, 16:02
общий
это ответ
Здравствуйте, Анна!
Для последовательности a[sub]n[/sub], n[$8805$]0 производящая функция имеет вид
В данном случае для a[sub]0[/sub] = -6, a[sub]1[/sub] = 2, a[sub]n[/sub] = 2a[sub]n-1[/sub]+15a[sub]n-2[/sub] имеем
откуда
и
- производящая функция. Чтобы построить явную формулу для последовательности, разложим знаменатель производящей функции на множители:
и разобьём её на сумму простых дробей:
Приравнивая коэффициенты в числителе, получаем 3A - 5B = 14 и A + B = -6, откуда A = -2, B = -4 и
Воспользовавшись свойствами геометрической прогрессии:
получим
откуда
Первые несколько элементов последовательности по этой формуле будут равны: a[sub]0[/sub] = -6, a[sub]1[/sub] = 2, a[sub]2[/sub] = -86, a[sub]3[/sub] = -142, a[sub]4[/sub] = -1574, a[sub]5[/sub] = -5278 и т.д.
Для последовательности a[sub]n[/sub], n[$8805$]0 производящая функция имеет вид
В данном случае для a[sub]0[/sub] = -6, a[sub]1[/sub] = 2, a[sub]n[/sub] = 2a[sub]n-1[/sub]+15a[sub]n-2[/sub] имеем
откуда
и
- производящая функция. Чтобы построить явную формулу для последовательности, разложим знаменатель производящей функции на множители:
и разобьём её на сумму простых дробей:
Приравнивая коэффициенты в числителе, получаем 3A - 5B = 14 и A + B = -6, откуда A = -2, B = -4 и
Воспользовавшись свойствами геометрической прогрессии:
получим
откуда
Первые несколько элементов последовательности по этой формуле будут равны: a[sub]0[/sub] = -6, a[sub]1[/sub] = 2, a[sub]2[/sub] = -86, a[sub]3[/sub] = -142, a[sub]4[/sub] = -1574, a[sub]5[/sub] = -5278 и т.д.
Форма ответа
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