How to find upper and lower bound without using formula
Another approach:


$n^2/4=(n/2)^2= \underbrace{n/2 + n/2 +\cdots + n/2}_{n/2 \text{ times}} \le 1+2+\cdots + n \le \underbrace{n + n +\cdots + n}_{n \text{ times}} = n^2 $ 
$n^2= \underbrace{n + n +\cdots + n}_{n \text{ times}} \le n+(n+1)+\cdots + 2n \le \underbrace{2n + 2n +\cdots + 2n}_{n \text{ times}} = 2n^2 $