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Total distance = S 

time taken for the first half of S :  [ S / 2 ] / V₀  =>  t1 = S/(2V₀)

time taken for the second part = 1/4 th of S :    [ S / 4 ] / V₁  =>  t2 = S/(4 V₁)

time taken for the third part :  t3 =  [S/4] / V₂ = S/(4V₂)

Total duration = T = S/4 * [2/V₀ + 1/V₁  + 1/V₂ ]

   =>  Average velocity = avg speed = S / T = 4 / [2/V₀ + 1/V₁ + 1/V₂ ]

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it is mentioned that the time intervals for the second half distance are equal and not the distances as you have mentioned in your answer . can you please check and update your answer accordingly !!! Thank You!!!
you are right.
  • Brainly User
       Velocity of Half of distance traveled = V0
   Let D1 be the distance traveled with a velocity of V1 and T be the time taken to travel this distance  and D2 be the distance traveled with the velocity of V1 and T be the time taken to travel this distance.
     therefore ,
             V1 =D1/T   and V2 = D2/T
    or D1 = V1T      and D2 = V2T
    Hence total distance Traveled = D1 + D2 = V1T + V2T  = T (V1+V2)
   also total time = T+T  = 2T
         Therefore ,
            Average Velocity of Second half of Journey  Vr= total distance / total time 
                                                                           Vr= T(V1+V2)/2T = V1+V2/T
   Since body travels half of the distance with a velocity V0 and The remaining half with a velocity (Average Velocity Vr) , therefore the average velocity of the body is given as
             Vavg = 2V0Vr/V0+Vr
                     =2V0(V1+V2)/2 / V0+(V1+V2/2)
                     =2V0V1+2V0V2 / 2V0+V1+V2

Hope this helps you!!!
Thanks !!!