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Консультация № 168783
02.06.2009, 17:11
0.00 руб.
0
1
1
Здравствуйте, уважаемые эксперты. Помогите, пожалуйста, решить задачу:
http://webfile.ru/3668667
http://webfile.ru/3668667
Обсуждение
03.06.2009, 22:31
общий
это ответ
Здравствуйте, Болдырев Тимофей.
В соответствии с формулой бинома Ньютона,
(1 + x)n = Cn0 + Cn1x + Cn2x2 + … + Cnnxn. (1)
Дифференцируя выражение (1), получаем
((1 + x)n)’ = Cn1 + 2xCn2 + … + nxn – 1Cnn. (2)
С другой стороны,
((1 + x)n)’ = n(1 + x)n – 1. (3)
Приравнивая выражения (2) и (3) и полагая x = 1, доказываем требуемое:
Cn1 + 2Cn2 + … + nCnn = 2n – 1n.
С уважением.
В соответствии с формулой бинома Ньютона,
(1 + x)n = Cn0 + Cn1x + Cn2x2 + … + Cnnxn. (1)
Дифференцируя выражение (1), получаем
((1 + x)n)’ = Cn1 + 2xCn2 + … + nxn – 1Cnn. (2)
С другой стороны,
((1 + x)n)’ = n(1 + x)n – 1. (3)
Приравнивая выражения (2) и (3) и полагая x = 1, доказываем требуемое:
Cn1 + 2Cn2 + … + nCnn = 2n – 1n.
С уважением.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
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