Here is a small challenge for your entertainment and gratification. For any natural number $n$, let $f(n)$ denote the sum of the numbers from 1 to $n$. Thus $f(1) = 1$, $f(2) = 1 + 2 = 3$, $f(3) = 1 + 2 + 3=6$, $f(100) = 1 + 2 + 3 + \ldots + 100 = 5050$, etc.
It turns out that $f$ is a polynomial of degree 2 in $n$. Figure out the coefficients of $f$:
$f(n)=$ $n^2$ $+$ $n$ + ,
There is a story about Carl Friedrich Gauss (1777-1855) who may have been the most outstanding mathematician in human history. According to the story, when Gauss was seven years old, his teacher at one stage was unhappy with the class and as a punishment he asked them to compute $f(100)$. Gauss' class mates started writing the numbers from 1 to 100 on their paper, and adding those numbers. Gauss stared at the ceiling and then wrote the single number $5050$ on the sheet and handed it in. You aren't Gauss, but you also aren't seven years old, so maybe you can figure out what he was thinking!
Hint: Think about how to do this in your head for large values of $n$.