Let's call the sum we are looking for $S_n$. The idea is to rewrite $S_n$ as the sum from $n$ to $1$ and then add both equations. Each element of the sum will nicely pair up:

Which leads to: $S_n=\frac{n\left(n+1\right)}{2}$. It is believed that Gauss discovered this method when he was just 9 years old.

If you have some intuition about the result or want to prove that Gauss found the right answer, you can prove that the result is right by induction.

Let's prove that the solution is correct for $n=1$:

And then let's prove that $S_n$ is correct if $S_{n-1}$ is correct:

We can express the idea behind the first method above by using a formal notation. We need to use 4 properties:

We then have:

And we can add the first and last sum:

Let's define the sum of the square of every number from $1$ to $n$:

We then have:

Since the last sum is 0 for $k=1$, we can write:

We can cancel the ${\displaystyle \sum _{k=1}^n}{k}^2$:

Which leads to: $S_n=\frac{n\left(n+1\right)}{2}$

The repertoire method helps solve general forms of equations, e.g.:

This method works when the solution has the form:

We first set $S_n=1$:

We then set $S_n=n$:

Finally, we set $S_n=n^2$:

Which results in the general solution:

Links

• Concrete Mathematics by Ronald L. Graham, Donald E. Knuth and Oren Patashnik
• Carl Friedrich Gauss (wikipedia.org)
• The repertoire method in "Concrete Mathematics"