South African Mathematics Olympiad Third Round

Two sequences of real numbers are defined as follows: $$u_1=0,u_{n+1}=\frac{1}{2}(u_n+v_n)$$ $$v_1=1,v_{n+1}=\frac{1}{4}(u_n+3v_n)$$ Find the value of $v_{2016}-u_{2016}$.