Does the mean value theorem for integrals give average velocity?

[Ali Asadullah]

In the photo attached, i have tried to derive the formula that average velocty is equal to the sum of initial and final velocities divided by two. But i am not sure about one step encircled in the photo. I don't know whether encircled step is right or wrong. I have another idea of deriving this formula which is similar to this but skips this step.
I assumed that we calculate velocities after each one second, but if we suppose that we somehow calculate velocity after each instant then sum of velocities will gives us the integral of velocity function and dividing it by "n" will give us average velocity according to Mean Value Theorem for Integrals.
My question is whether my second appraoch is right?

 

[Chestermiller]

Average velocity is equal to the vector difference in displacement divided by the time interval