Witty functional equation
Let $$f(x) = 2/(4^x + 2)$$ for real numbers $x$. Evaluate $$f(1/2001) + f(2/2001) + f(3/2001) + \cdots + f(2001/2001)$$
Any idea?
We have $\displaystyle f(1-x)=\frac{2}{4^{1-x}+2}=\frac{2\cdot 4^x}{4+2\cdot 4^x}=\frac{4^x}{2+ 4^x}$
$\displaystyle f(x)+f(1-x)=\frac{2}{4^x+2}+\frac{4^x}{2+ 4^x}=1$
The required sum is
$$\frac{2000}{2}\times 1+f(1)=1000+\frac{2}{4+2}=\frac{3001}{3}$$