用地铁线路理解单重求和与双重求和的计算

一、前言

用求和符号 $\sum$ 表示的求和运算是一种非常基本运算形式。在本文中，「荒原之梦考研数学」将通过地铁线路的方式，为同学们形象地解释单重求和与双重求和的计算思路。

二、正文

单重求和

对于求和运算 $\sum_{i=1}^{4} (i)$, 我们有：

$$
\begin{aligned}
& \sum_{i=1}^{4} (i) \\ \\
= \ & \sum_{i=1}^{4} (1) + \sum_{i=1}^{4} (2) + \sum_{i=1}^{4} (3) + \sum_{i=1}^{4} (4) \\ \\
= \ & 1 + 2 + 3 + 4 \\ \\
= \ & \textcolor{springgreen}{10}
\end{aligned}
$$

如果用地铁线路表示求和运算 $\sum_{i=1}^{4} (i)$, 则如图 01 所示：

当然，被求和的式子 $(i)$ 可以有多种具体形式，例如，对于求和运算 $\sum_{i=1}^{3} (i \textcolor{lightgreen}{+1})$, 我们有：

$$
\begin{aligned}
& \sum_{i=1}^{3} (i \textcolor{lightgreen}{+1}) \\ \\
= \ & (1 \textcolor{lightgreen}{+1}) + (2 \textcolor{lightgreen}{+1}) + (3 \textcolor{lightgreen}{+1}) \\ \\
= \ & \textcolor{springgreen}{9}
\end{aligned}
$$

双重求和

如果说单重求和就是遍历地铁站，并将每座地铁站点对应的数值相加求和，那么，双重求和就是不仅要遍历地铁站，还要同时遍历每座地铁站的进出站口（只需要将每个进出站口对应的数值相加），如图 02 所示：

于是可知，对于求和运算 $\sum_{j=1}^{3} \sum_{i=1}^{4} (i+j)$, 我们有：

$$
\begin{aligned}
& \sum_{j=1}^{3} \sum_{i=1}^{4} (i+j) = \sum_{\textcolor{lightgreen}{j}=1}^{3} \left[ \sum_{\textcolor{orange}{i}=1}^{4} (\textcolor{orange}{i} + \textcolor{lightgreen}{j}) \right] \\ \\
= \ & \sum_{j=1}^{3} \left[ \sum_{\textcolor{orange}{i}} (\textcolor{orange}{1} + j) \right] + \sum_{j=1}^{3} \left[ \sum_{\textcolor{orange}{i}} (\textcolor{orange}{2} + j) \right] \\
+ \ & \sum_{j=1}^{3} \left[ \sum_{\textcolor{orange}{i}} (\textcolor{orange}{3} + j) \right] + \sum_{j=1}^{3} \left[ \sum_{\textcolor{orange}{i}} (\textcolor{orange}{4} + j) \right] \\ \\
= \ & \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (1+\textcolor{lightgreen}{1}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (1+\textcolor{lightgreen}{2}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (1+\textcolor{lightgreen}{3}) \right] \\
+ \ & \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (2+\textcolor{lightgreen}{1}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (2+\textcolor{lightgreen}{2}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (2+\textcolor{lightgreen}{3}) \right] \\
+ \ & \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (3+\textcolor{lightgreen}{1}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (3+\textcolor{lightgreen}{2}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (3+\textcolor{lightgreen}{3}) \right] \\
+ \ & \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (4+\textcolor{lightgreen}{1}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (4+\textcolor{lightgreen}{2}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (4+\textcolor{lightgreen}{3}) \right] \\ \\
= \ & (1+1) + (1+2) + (1+3) \\
+ \ & (2+1) + (2+2) + (2+3) \\
+ \ & (3+1) + (3+2) + (3+3) \\
+ \ & (4+1) + (4+2) + (4+3) \\ \\
= \ & 2 + 3 + 4 \\
+ \ & 3 + 4 + 5 \\
+ \ & 4 + 5 + 6 \\
+ \ & 5 + 6 + 7 \\ \\
= \ & \textcolor{springgreen}{54}
\end{aligned}
$$

类似地，对于求和运算 $\sum_{j=1}^{3} \sum_{i=1}^{4} (i \times j)$, 我们有：

$$
\begin{aligned}
& \sum_{j=1}^{3} \sum_{i=1}^{4} (i \times j) = \sum_{\textcolor{lightgreen}{j}=1}^{3} \left[ \sum_{\textcolor{orange}{i}=1}^{4} (\textcolor{orange}{i} \times \textcolor{lightgreen}{j}) \right] \\ \\
= \ & \sum_{j=1}^{3} \left[ \sum_{\textcolor{orange}{i}} (\textcolor{orange}{1} \times j) \right] + \sum_{j=1}^{3} \left[ \sum_{\textcolor{orange}{i}} (\textcolor{orange}{2} \times j) \right] \\
+ \ & \sum_{j=1}^{3} \left[ \sum_{\textcolor{orange}{i}} (\textcolor{orange}{3} \times j) \right] + \sum_{j=1}^{3} \left[ \sum_{\textcolor{orange}{i}} (\textcolor{orange}{4} \times j) \right] \\ \\
= \ & \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (1 \times \textcolor{lightgreen}{1}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (1 \times \textcolor{lightgreen}{2}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (1 \times \textcolor{lightgreen}{3}) \right] \\
+ \ & \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (2 \times \textcolor{lightgreen}{1}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (2 \times \textcolor{lightgreen}{2}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (2 \times \textcolor{lightgreen}{3}) \right] \\
+ \ & \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (3 \times \textcolor{lightgreen}{1}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (3 \times \textcolor{lightgreen}{2}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (3 \times \textcolor{lightgreen}{3}) \right] \\
+ \ & \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (4 \times \textcolor{lightgreen}{1}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (4 \times \textcolor{lightgreen}{2}) \right] + \sum_{\textcolor{lightgreen}{j}} \left[ \sum_{i} (4 \times \textcolor{lightgreen}{3}) \right] \\ \\
= \ & (1 \times 1) + (1 \times 2) + (1 \times 3) \\
+ \ & (2 \times 1) + (2 \times 2) + (2 \times 3) \\
+ \ & (3 \times 1) + (3 \times 2) + (3 \times 3) \\
+ \ & (4 \times 1) + (4 \times 2) + (4 \times 3) \\ \\
= \ & 1 + 2 + 3 \\
+ \ & 2 + 4 + 6 \\
+ \ & 3 + 6 + 9 \\
+ \ & 4 + 8 + 12 \\ \\
= \ & \textcolor{springgreen}{60}
\end{aligned}
$$

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