Gauss’s most famous story is from when he was about 9 years old. To keep the class busy, the Professor Büttner told the students to add up all the numbers from 1 to 100. In a few seconds, Gauss placed the blackboard on the table with the correct answer: 5050. (The story was told by the first biographer of Gauss, and the exact numbers in the account may have been different, but the idea is the same.)

The trick: combine the first with the last, the second with the penultimate, and so on.

$$\begin{aligned} S &= 1 + 2 + 3 + \cdots + 98 + 99 + 100 \\ S &= 100 + 99 + 98 + \cdots + 3 + 2 + 1 \\ 2S &= \underbrace{101 + 101 + 101 + \cdots + 101 + 101 + 101}_{100 \text{ vezes}} = 10\,100 \\ S &= 5050. \end{aligned}$$

For any $n$, the same idea gives

$$1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}.$$

The figure shows why it works: the sum is a ladder, and two equal ladders form a rectangle.

A staircase with 1 + 2 + … + n squares. Place one copy face down and count the rectangle.

The numbers $1, 3, 6, 10, 15, 21, \ldots$ that appear as stairs are the triangular numbers. Already we talk about them and PA on this site: see the triangular numbers and arithmetic progression.

All topics by Carl Friedrich Gauss · ← The life of Carl Friedrich Gauss · The polygon of 17 sides →

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