Overview
The story goes that a young Gauss, told to add 1 to 100 as busywork, saw the answer almost instantly. The trick is one of the most reused ideas in mathematics.
How to solve Gauss's Sum
- Pair the smallest and largest terms: 1 + 100 = 101.
- Every inward pair (2+99, 3+98, …) also totals 101, and there are 50 such pairs.
- So the sum is 50 × 101 = 5050.
The key insight
Reversing a sum and adding it to itself makes every column identical. That symmetry gives the closed form n(n+1)/2 for any n.
Variations & echoes
- The same pairing sums any arithmetic series.
- It's the basis of the triangular numbers 1, 3, 6, 10, ….
Frequently asked questions
Does it work for odd counts?
Yes. n(n+1)/2 always gives a whole number because one of n or n+1 is even.