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

Question

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

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

have you heard arithmetic progression?
https://en.wikipedia.org/wiki/Arithmetic_progression — duanduan, Dec 24 '16 at 23:28
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 :http://oi67.tinypic.com/e9vvi8.jpg — Evariste, Dec 25 '16 at 00:28

score 7 · Accepted Answer · answered Dec 24 '16 at 23:30

7

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

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

answered Dec 24 '16 at 23:30

Brian M. Scott

616,228

2

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

score 3 · Answer 2 · edited Dec 24 '16 at 23:34

3

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

edited Dec 24 '16 at 23:34

Ben Grossmann

225,327

answered Dec 24 '16 at 23:30

Asinomás

105,651

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

score 2 · Answer 3 · answered Dec 24 '16 at 23:29

2

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

You can show by induction.

answered Dec 24 '16 at 23:29

Zaid Alyafeai

14,343

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

3 Answers3