Originally Posted by
Soroban Hello, mich13!
The symbol $\displaystyle {\bf{\color{blue}>}}$ means "greater than".
You may be expected to know this formula . . .
The sum of the positive integers from $\displaystyle 1$ to $\displaystyle n$ is: .$\displaystyle \frac{n(n+1)}{2}$
So we have: .$\displaystyle \frac{n(n+1)}{2}\;> \;2007\quad\Rightarrow\quad n^2 + n \;>\;4014$
We have a quadratic inequality: .$\displaystyle n^2 + n - 4014\;>\:0$
The equation: .$\displaystyle n^2 + n - 4014 \:=\:0$
. . has roots: .$\displaystyle n \;=\;\frac{-1 \pm\sqrt{1^2 - 4(1)(4014)}}{2(1)}\;=\;\frac{-1\pm\sqrt{16057}}{2} \;=\;\begin{Bmatrix}\;62.658... \\ \text{-}64.739...\end{Bmatrix}$
Therefore: . $\displaystyle \boxed{n\:\geq\:63}$