Question

Average velocity = initial velocity + final velodrome / 2.
Average velocity = u+v/2.
Distance traveled = AV × T.
S=(u+v/2)t.
S=(u+u+at/2)t.
Because 1 equation of motion v=u+at/2.
S=(2u+at/2)t
S=(2ut+at^/2).
S=(2ut/2 + at^/2).
S=ut+1/2at^.
Hence proved

Answer 1

I noticed that you want to clarity in 
           if average velocity = (final velocity+initial velocity)/2,then  prove that 
S = ut + 1/2at^2 
where, u is initial velocity , v is final velocity and a is acceleration of the body.

solution:- total distance = average velocity*time taken
Also, we know,
v = u + at 
use this in eqns in above 
total distance = (v+u)/2 *t 
                       = (u +at +u)/2 *t
                      = (2u+at)t/2
                      =2ut/2 + at^2/2
                      = ut + 1/2at^2 
                              hence proved//

Answer 2

Answer: YOUR ANSWER

Explanation: I noticed that you want to clarity in  

          if average velocity = (final velocity+initial velocity)/2,then  prove that  

S = ut + 1/2at^2  

where, u is initial velocity , v is final velocity and a is acceleration of the body.

solution:- total distance = average velocity*time taken

Also, we know,

v = u + at  

use this in eqns in above  

total distance = (v+u)/2 *t  

                      = (u +at +u)/2 *t

                     = (2u+at)t/2

                     =2ut/2 + at^2/2

                     = ut + 1/2at^2  

                             hence proved//