Cho hàm số: $\Large f(x)=\dfrac{1}{2}\log_{2}\left(\dfrac{2x}{1-x}\rig

Cho hàm số: $\Large f(x)=\dfrac{1}{2}\log_{2}\left(\dfrac{2x}{1-x}\right)$. Tính tổng: 

$\Large S=f\left(\dfrac{1}{2017}\right)+f\left(\dfrac{2}{2017}\right)+f\left(\dfrac{3}{2017}\right)+...+f\left(\dfrac{2015}{2017}\right)+f\left(\dfrac{2016}{2017}\right)$

 

• A.

  A. 2017

• B.

  B. 2016

• C.

  C. 4032

• D.

  D. 1008

Chọn D

Xét: $\Large f(x)+f(1-x)=\dfrac{1}{2}.\log_{2}\left(\dfrac{2x}{1-x}\right)+\dfrac{1}{2}\log_{2}\left(\dfrac{2(1-x)}{x}\right)=\dfrac{1}{2}\log_{2}4=1$

Do đó: $\Large f\left(\dfrac{1}{2017}\right)+f\left(\dfrac{2016}{2017}\right)=1, f\left(\dfrac{2}{2017}\right)+f\left(\dfrac{2015}{2017}\right)=1, ...$

Suy ra: $\Large S=1+1+...+1+1=1008$