Gauss's sum — the trick that changes how you see arithmetic

Fri, 20 Mar 2026 00:00:00 +0000

Problem: Find the sum of the first $n$ natural numbers: $1 + 2 + 3 + \cdots + n$.

Solution:

Write the sum twice — once forwards, once backwards, and add column by column:

$$S = 1 + 2 + 3 + \cdots + n$$ $$S = n + (n-1) + (n-2) + \cdots + 1$$ $$2S = (n+1) + (n+1) + \cdots + (n+1) \quad (n \text{ times})$$

Therefore:

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