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A body travelling along a straight line travels one half of the total didtance with the velocity v0the remaining part of distance was covered by the

velocity of v1 for half of the time and with velocit v2 for other half of the time. find the average velocity



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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.
       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 !!!
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