Use the approach in Gauss's Problem to find the following

Consider the series ${\displaystyle 1+2+3+4+\dots +1001}$.
For the series ${\displaystyle 1+2+3+4+\dots +1001}$, it is clear that there are ${\displaystyle 1001}$ terms, which is odd.
So, add the series to itself in reverse order.
Note that each pair has a sum of ${\displaystyle 1002}$ and there are ${\displaystyle 1001}$ pairs which gives:
${\displaystyle 1002\left(1001\right)=1003002}$
Since this is the sum of the series added to itself, the sum of the original series is half this sum.
The sum of ${\displaystyle 1+2+3+4+\dots +1001}$ is then ${\displaystyle \frac{1}{2}\left(1003002\right)=501501}$
This is the answer.