Question 1010816
The expression has {{{100}}} terms of the form {{{(x-n/2)}}} , with alternating {{{"+"}}} and {{{"-"}}} signs.
On removing the brackets, the {{{"+ x"}}} and {{{"- x"}}} terms cancel out.
After taking out the common factor {{{1/2)}}} ,
we get a sum of {{{100}}} integer terms that can be grouped in {{{100/2=50}}} pairs of the form {{{(m+1) - m=1}}} .
{{{(x-1/2) - (x-2/2) + (x-3/2) -(x-4/2)+ "..."+(x-99/2) - (x-100/2)}}}
={{{x-1/2-x+2/2+x-3/2-x+4/2+ "..."+x-99/2-x+100/2}}}
={{{-1/2+2/2-3/2+4/2-"..."-99/2+100/2}}}
={{{(1/2)*(-1+2-3+4-"..."-99+100)}}}
={{{(1/2)*((2-1)+(4-3)+"..."+(100-99))}}}
={{{(1/2)*50=highlight(25)}}}