SOLUTION: (x-1/2) - (x-2/2) + (x-3/2) - ... - (x-100/2) Please have clear steps and working out. Thanks!

Algebra ->  Sequences-and-series -> SOLUTION: (x-1/2) - (x-2/2) + (x-3/2) - ... - (x-100/2) Please have clear steps and working out. Thanks!      Log On


   



Question 1010816: (x-1/2) - (x-2/2) + (x-3/2) - ... - (x-100/2)
Please have clear steps and working out. Thanks!

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The expression has 100 terms of the form %28x-n%2F2%29 , with alternating %22%2B%22 and %22-%22 signs.
On removing the brackets, the %22%2B+x%22 and %22-+x%22 terms cancel out.
After taking out the common factor 1%2F2%29 ,
we get a sum of 100 integer terms that can be grouped in 100%2F2=50 pairs of the form %28m%2B1%29+-+m=1 .

=x-1%2F2-x%2B2%2F2%2Bx-3%2F2-x%2B4%2F2%2B+%22...%22%2Bx-99%2F2-x%2B100%2F2
=-1%2F2%2B2%2F2-3%2F2%2B4%2F2-%22...%22-99%2F2%2B100%2F2
=%281%2F2%29%2A%28-1%2B2-3%2B4-%22...%22-99%2B100%29
=%281%2F2%29%2A%28%282-1%29%2B%284-3%29%2B%22...%22%2B%28100-99%29%29
=%281%2F2%29%2A50=highlight%2825%29