Question

求 $\sum_{i=1}^{100}\sum_{j=1}^{100}(i+j)?$

What is $\sum_{i=1}^{100}\sum_{j=1}^{100}(i+j)?$

$100{,}100$

$500{,}500$

$505{,}000$

$1{,}001{,}000$

$1{,}010{,}000$

答案：E

知识点：求和等差数列

难度评级：1620

解答：

拆开求和： $\begin{gathered} \sum_{i=1}^{100}\sum_{j=1}^{100}(i+j) \\ =\sum_{i=1}^{100}\sum_{j=1}^{100}i \\ {}+\sum_{i=1}^{100}\sum_{j=1}^{100}j \\ =100\sum_{i=1}^{100}i \\ {}+100\sum_{j=1}^{100}j. \end{gathered}$

因为 $\sum_{k=1}^{100}k=5050$ ，结果为 $100\cdot5050$ $+100\cdot5050$ $=1{,}010{,}000$ 。

所以正确答案是 E。

Splitting the sum, $\begin{gathered} \sum_{i=1}^{100}\sum_{j=1}^{100}(i+j) \\ =\sum_{i=1}^{100}\sum_{j=1}^{100}i \\ {}+\sum_{i=1}^{100}\sum_{j=1}^{100}j \\ =100\sum_{i=1}^{100}i \\ {}+100\sum_{j=1}^{100}j. \end{gathered}$

Since $\sum_{k=1}^{100}k=5050,$ this equals $100\cdot5050$ $+100\cdot5050$ $=1{,}010{,}000.$

Thus, the correct answer is E.

2018 AMC 12B 第 9 题

答案：E

难度评级：1620

解答：

其他年份的第 9 题