Answer:
1007
Explanation:
Divide the expression by two parts: the first one with all positive integer (the integer which can't be divided by two), 1+3+5+...2013 and the rest -2-4-6-...-2012
If you notice both parts are arithmetical sequences
Explore the first one 1+3+5...+2013
a1=1 d=3-1=2 an=2013
a1+d(n-1)=an
1+2(n-1)=2013
2(n-1)=2012
n-1=1006
n=1007
Sn=( 1/2)*(a1+an)*n= 0.5*2014*1007=1007*1007=1014049
-2+(-4)-...-2012
a1=-2
d=a2-a1= -4-(-2)=-2
a1+d(n-1)=an
-2+(-2)(n-1)=-2012
n=1006
Sn= (a1+an)/2*n=(-2-2012)/2*1006=-1007*1006
-1007*1006+1007*1007= 1007- the answer
The second mean
1-2=-1
3-4=-1
5-6=-1
There are 2012/2=1006 couples and every of them has sum -1
-1*1006=-1006
-1006+2013=1007