Suppose you have a list of numbers from zero to one hundred. How quickly can you add them all up without using a calculator?
You can use Gauss method
$\displaystyle S=1 + 2 + 3 + \cdots + 98 + 99 + 100$
$\displaystyle S=100 + 99 + 98 + \cdots + 3 + 2 + 1$
$\displaystyle 2S=101 + 101 + 101 + \cdots + 101 + 101 + 101$
2S is the sum of 100 terms equal to 101
Therefore $\displaystyle S=\frac{100\cdot101}{2}=5050$
See also this thread.