A particle covers half of its total distance with speed $v_1$ and the rest half distance with speed $v_2$. Its average speed during the complete journey is 

 Let S be the total distance travelled by the particle. Let $t_1$ be the time taken by the particle to cover first half of the distance. Then $t_1=\frac{S/2}{v_1}=\frac{S}{2v_1}$ Let $t_2$ be the time taken by the particle to cover remaining half of the distance. Then  $t_2=\frac{S/2}{v_2}=\frac{S}{2v_2}$

Average speed,

$v_{\mathrm{av}}=\frac{\text{ Total distance travelled }}{\text{ Total time taken }}$

$=\frac{S}{t_1+t_2}=\frac{S}{\frac{S}{2v_1}+\frac{S}{2v_2}}=\frac{2v_1{v}_2}{v_1+v_2}$