# Why does $1+2+\dots+2003=\dfrac{2004\cdot2003}2$? [duplicate]

Why does $1+2+\dots+2003=\dfrac{2004\cdot2003}2$?

Sorry if this is missing context; not really much to add...

## marked as duplicate by Stefan Perko, suomynonA, Community♦Dec 24 '16 at 23:44

• have you heard arithmetic progression? en.wikipedia.org/wiki/Arithmetic_progression – duanduan Dec 24 '16 at 23:28
• Because half a square is a triangle. – Evariste Dec 24 '16 at 23:31
• @Evariste what? – suomynonA Dec 24 '16 at 23:32
• @Evariste isn't it a rectangle? – Jorge Fernández Hidalgo Dec 25 '16 at 0:17
• I realize I was a bit quick, but essentially the idea is a square. As you can see, $2003\times 2004$ is very close to a square. This pictures helps the understanding a bit :oi67.tinypic.com/e9vvi8.jpg – Evariste Dec 25 '16 at 0:28

$$\begin{array}{ccc} S&=&1&+&2&+&3&+&\ldots&+&2001&+&2002&+&2003\\ S&=&2003&+&2002&+&2001&+&\ldots&+&3&+&2&+&1\\ \hline 2S&=&2004&+&2004&+&2004&+&\ldots&+&2004&+&2004&+&2004 \end{array}$$

There are $2003$ columns, so $2S=2003\cdot2004$, and therefore $S=\dfrac{2003\cdot2004}2$.

• This formatting is gorgeous; great use of array. – Omnomnomnom Dec 24 '16 at 23:33
• @suomynonA: You’re welcome! – Brian M. Scott Dec 24 '16 at 23:34

By symmetry, the numbers are all centered around $\frac{n+1}{2}$, and there are $n$ of them.

• Yeah, the number $\frac{n+1}{2}$ is an axis of symmetry for the numbers, when you look at them on the number line. – Jorge Fernández Hidalgo Dec 24 '16 at 23:32
• This kind of answer is only useful to those who are used to phrases like "by symmetry" – Omnomnomnom Dec 24 '16 at 23:32
• well. I tried to answer the question "why" as much as possible. – Jorge Fernández Hidalgo Dec 24 '16 at 23:33
• Oh, I see, thanks – suomynonA Dec 24 '16 at 23:33

$$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$

You can show by induction.