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«Статья по математике»
Сумма степеней n натуральных чисел.
Задача 1. Найти сумму: 1 + 2 + … + n.
Мы знаем формулу квадрата суммы
Это тождество справедливо для любого n. Распишем формулу в несколько строчек при различных n, начиная с 0, получим:
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_2.png)
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_3.png)
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_4.png)
………………………………….
………………………………….
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_5.png)
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_6.png)
Складываем ![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_7.png)
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_8.png)
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_9.png)
Обозначим
, получили формулу расчета ![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_11.png)
Докажем справедливость полученной формулы методом математической индукции
1. Проверим справедливость при n = 1.
- справедлива 1 = 1.
2. Предположим, что формула справедлива при всех значения до n, то есть равенство
верно и проверим верность для n+1.
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_14.png)
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_15.png)
Получили верное утверждение для любого n.
Задача 2. Найти сумму: ![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_16.png)
Для решения этой задачи будем использовать формулу куба суммы.
.
Будем выполнять точно такой же алгоритм вывода формулы:
0, ![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_18.png)
1, ![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_19.png)
2, ![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_20.png)
……………………………………………….
……………………………………………….
n-1, ![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_21.png)
n,
Складываем:
![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_23.png)
Введем обозначения: ![](https://fsd.multiurok.ru/html/2018/04/29/s_5ae5bb48be6a0/891266_24.png)
Получим:
![](data:image/png;base64,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)
;
![](data:image/png;base64,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)
Докажем, аналогично, как и в первой задаче методом математической индукции:
1. n = 1
верное.
2. Справедливо при n ![](data:image/png;base64,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)
3. Доказываем справедливость при n+1.
Формула справедлива.
.
Задача 3. Найти сумму:
.
Все повторяем снова, меняется только формула. ![](data:image/png;base64,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)
0, ![](data:image/png;base64,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)
1, ![](data:image/png;base64,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)
……………………………………………………..
n, ![](data:image/png;base64,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)
n+1, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHcAAAARCAIAAABo9qSaAAADLUlEQVR4nO1YO08iURh1FS0MGuOLRdFCxScFgUItTDZUNsSYoNHSlkqz0dLChkJ+A4kxxpJAQQIUNBRokAbdBA1oYaLiIyoJiiayZ7i7N5PhzjCMqIl6qjv3zvm+j/M95gZVPp+v+sYbQ/XRAXwJVFLl09NTnU5XQYOfBv9Ufnp6qqurU8CnxMfHx9XVVZfLVa6F1+QmkUg8PDwYjcYPocsHp/Le3p5er1em8s7OzsTEBBbBYHB4ePj5+bm2tlY+XXFuCJqbmwOBwM8CFNPb29s7OjqUBSATnMrHx8dms7ks2sXFxdTUVCQSyeVy2Wy2vr6+oaHh5OTk/Py8q6tLvh05ubm6uurr67u9vS0+UqvV2FepROfe3d3dwsICpNRqtUdHR4LTtra2l5cXaYklvIuBikN3uPgODw9LMtFcAwMDZH19fb2ysrK7u4s1IsDi13/ID4WAmRu+LyAUCt3f3zPp+CUajQb01tZWJh1xdnZ2ptPp+fn5Yvr6+npTUxN0FKNLe2dS+OJQCKvAYrGgjy4vLxcXF6enp+n+0NAQ0k7WLS0tGxsbm5ubWCPKs7Mz6SAk7DNzw/eFHwCZJOi4iVZXV4vRvV5vLBZDq3k8nmL68vJycYTyvZcUh4JTmd+ts7Ozbrfbbrdvb2/zDYkBPYvsMY/QpOgdsh4ZGYnH4wrs+3y+paUlsfB+FCBBR2yjo6M3NzdbW1vo4sp6L0mnEKp8cHAwMzODUYtxicdkMokPIzkiJYPp1tvbS9/Hm2IjlVnjAvt8MH39LoDsoF7KpaP0wuFwJpOxWq1QubLeS4pDwanMv13A0NzcHDI2NjaGR3BILxA3xXzBTOSDWcsC+3wwfQl2yqWPj4/jBoWOxmWm4t5LikPBqYxPEH2GocHBwT8F2Gw2JieVSk1OTmLR39+/trbGzF6VeC2XtC+BcukOhwOvoRScTuf7eOeL09jYGI1Gq4jKGNj0JVJ9mEcCMj9XPT099Fri9/tNJpP8QMXsi/kS7JRLhyj7+/tv7V1MHApOZYPBgK8EX2v5QLpqamoUEL8UOJW7u7sV8zH4KhfMp8X3f3LvgW+V3wN/AaORdgOCYJdFAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
=
.
Аналогично доказываем справедливость полученного результата.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEQAAAAZCAIAAAAHYR/MAAAC/UlEQVR4nO1YTUsyURSeVFy4sUwTaxN+JGJk/gFtFe3cJwSK0KLC4iXBH2AbFwZtXAUlhbaLSKSN0KqsRVBCKbUQS8VCaRMk6fvwTtwmZ6wpR4N4n4WcmXPPx3PPvefeUdJoNKjfAslPJyAkBCNzdXX19PQ0Pj7+g37aJVMqlRwOx9HRkUKhODg4GBgYGBwcpFWhUGh6eprWfmq+urq6uLiIN2w//NEWmYeHB5/Pl0qlIKtUqnq9TjLY3NyUyWRES4CJNxqNbPO+vr7t7W2Qb/LTPTL9/f0bGxuRSARyMBjs7e29v79XKpV4XF9fx/vZ2VlaS2AymZAr29xut7tcLpBp8tM9MkwsLy8zHy8uLtRqNX9zlAImbD9fQqe62ePjo1QqJY/X19cGg4GWRSIRfrPZrE6nIwMwGCZtBu0UGblc/vz8TPggb3p1gQlZZkxgMEzaDPpGZmdnx+/35/N5vV6fTqf5GN/c3ExNTUEYGRlJJBJarZaozGbzyckJtgGnlm2OPTY6Ovql1Ekn5CAzNzeHkPAYDod5ukN+mUyGU+V2u3O5HFvLLAvTfGtry+Px8IxLve+EBG9khoaGYrEYZtTr9fJ32gozMzNra2v8x1cqlfn5ef7jmZ3QZrMdHh5STDLxeBzHFtYYKj45OdlkrNFoUFZaBuHz8/NP4y0sLPBP7mMmH0enmVBMMmiO2DZ7e3uY1GKx2OSuUCjwz0xw8Iz+SmZlZQUTieaDSzQOY7w5OzuDbLVa6QGt5qanp0fgxN+DvtTzXBevZI6Pj3H01mq1sbExLMTb29vh4eH9/X1CptXcdOcLgh2d2QnR7i8vLylCZnd3l2mJ9TYxMcF5IHwbuKFgQ1arVUG8kU6IMjidTvolx6GJmkokEvRoi8UiSGAayWSy/TOeDZwo0WiUljnIlMvlu7u7l5cXsVgsVEhcltH6hfLGxOnpKZE5yOASHggEhA2Jvr+0tCSsTza69Nn85x+o1nczQdAlMh/fMoXC/z80vouOloX6ZZX5C4Z7fbbGRrEfAAAAAElFTkSuQmCC)
2.
=![](data:image/png;base64,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)
3.
Справедливость формулы доказана.
Задача 4. Найти сумму четвертых степеней натуральных чисел.
Из соотношения
можно получить сумму четвертых степеней по аналогии:
;
=![](data:image/png;base64,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)
Доказывается также, можно выполнить самостоятельно
Задача 4. Найти сумму пятых степеней натуральных чисел.
Все также как и выше.
![](data:image/png;base64,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)
;
![](data:image/png;base64,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)
Для других степеней подобные соотношения для нахождения суммы степеней похожие:
Во всех соотношениях используются коэффициенты из бинома Ньютона, которые определяются формулой:
.
В общем виде соотношение для вывода формулы суммы выглядит так:
В случае необходимости можно подобным образом находить суммы степеней n натуральных чисел.