f(x)=sin三分之πx,则f（1）+f(2)+f(3)...+f(2013)=?

Question

f(x)=sin三分之πx,则f（1）+f(2)+f(3)...+f(2013)=?

Answer 1

f(x)=sin(πx/3)
最小正周期是：2π/(π/3)=6
即：f(x)=f(x+6)
f(1)+f(2)+f(3)+f(4)+f(5)+f(6)
=sin(π/3)+sin(2π/3)+sin(π)+sin(4π/3)+sin(5π/3)+sin(2π)
=(√3)/2+(√3)/2+0-(√3)/2-(√3)/2+0
=0
因为：2013÷6=335余3
所以：
f(1)+f(2)+f(3)+……+f(2013)
=0×355+f(2011)+f(2012)+f(2013)
=f(1)+f(2)+f(3)
=sin(π/3)+sin(2π/3)+sin(π)
=(√3)/2+(√3)/2+0
=√3