find $f(\frac{1}{2014})+f(\frac{2}{2014})+.....+f(\frac{2013}{2014})$ of $f(x)=\frac{2}{2+4^x}$

Question

$f(x)=\frac{2}{2+4^x}$ find $f(\frac{1}{2014})+f(\frac{2}{2014})+.....+f(\frac{2013}{2014})$

Please guide me through it, the only step I know is probably to eliminate the denominator ps. Not a homework

Answer 1

HINT:

$$f(1-x)=\dfrac2{2+4^{1-x}}=\dfrac{2\cdot4^x}{2\cdot4^x+4}=\dfrac{4^x}{4^x+2}=\dfrac{4^x+2-2}{4^x+2}=1-f(x)$$

Set $x=\dfrac r{2014}, 1\le r\le2013$ and add