Question

Simplify:

(x-1/2)-(x-2/2)+(x-3/2)-...-(x-100/2)=

Answer 1

(x-1/2)-(x-2/2)+(x-3/2)-...-(x-100/2)
= x - 1/2 - x + 2/2 + x - 3/2 - .... - x + 100/2
= x-x+x-x+x-....-x -1/2 + 2/2 - 3/2 - ... + 100/2
looking at the last term, we see there are 100 terms, so the x's drop out
= -1/2 + 2/2 - 3/2 - ... + 100/2
= (-1/2)(1 - 2 + 3 - .... + 99 - 100)
= (-1/2)[ (1-2) + (3-4) + ... + (99-100]
= (-1/2)[ 50(-1)]
= 25

Not claiming that this is the shortest or best way of doing it, but it is easy to understand

Answer 2

(x-1/2)-(x-2/2)+(x-3/2)-...-(x-100/2)

x-x+x-x+...+x-x - 1/2 + 2/2 - 3/2 + ... - 99/2 + 100/2

the x-x+x-x terms cancel out, and we are left with

1/2 (-1+2 -3+4 ... -99+100)
= 1/2 (1+1...+1) (50 times)
= 1/2 * 50
= 25