f(m+n)=f(m)+f(n)+1/2,且f(1/2)=0.
f(1/2+1/2)=f(1/2)+f(1/2)+1/2=2f(1/2)+1/2=1/2.
∴f(1)=1/2.
f(3x-x^2-1)>1/2=f(1),
又 ∵当x>1/2时,f(x)>0,说明f(x)为增函数,
有(3X-X^2-1)>1,
X^2-3X+2<0,
1f(1)=1/2,
f(2)=f(1+1)=f(1)+f(1)+1/2=3*1/2=3/2,
f(3)=f(2+1)=f(2)+f(1)+1/2=3/2+1/2+1/2=5/2,
f(4)=f(3+1)=f(3)+f(1)+1/2=7/2
.
f(n)=(2n-1)/2.
∴f(1)+f(2)+.+f(n)=1/2+3/2+5/2+7/2+.+(2n-1)/2,
=[1/2+(2n-1)/2]*n/2
=n^2/2.