# What is (x+1)+(x-2)+(x+3)+(x-4)+...+(x+99)+(x-100) ?

Mar 11, 2017

$\left(x + 1\right) + \left(x - 2\right) + \left(x + 3\right) + \left(x - 4\right) + \ldots + \left(x + 99\right) + \left(x - 100\right) = 100 x - 50$

#### Explanation:

Since the signs alternate, it simplifies the problem if we group the expressions in pairs like this...

$\left(x + 1\right) + \left(x - 2\right) + \left(x + 3\right) + \left(x - 4\right) + \ldots + \left(x + 99\right) + \left(x - 100\right)$

$= \left(\left(x + 1\right) + \left(x - 2\right)\right) + \left(\left(x + 3\right) + \left(x - 4\right)\right) + \ldots + \left(\left(x + 99\right) + \left(x - 100\right)\right)$

$= {\overbrace{\left(2 x - 1\right) + \left(2 x - 1\right) + \ldots + \left(2 x - 1\right)}}^{\text{50 times}}$

$= 50 \left(2 x - 1\right)$

$= 100 x - 50$