Show that when a particle with uniform acceleration, the distances described in consecutive equal intervals of time are in A.P.

Explanation

Let us consider equal intervals of time,each having value t.Suppose u be the velocity at the beginnig of the first time interval and a the uniform acceleration.
Distance travelled during the first n intervals [i.e., In time (n-1)t]
`S_(1)=u(nt)+(1)/(2)(nt)^(2)a`
Distance travelled during the first (n-1) interval [i.e., in time (n-1)t]
`S_(2)=u(n-1)t+(1)/(2)a(n-1)^(2)t^(2)`
Distance travelled in the nth time interval=`S_(1)-S_(2)`
`=(unt+(1)/(2)an^(2)t^(2))-[unt-ut+(1)/(2)an^(2)t^(2)-ant^(2)+(1)/(2)at^(2)]`
`=ut+ant^(2)-(1)/(2)at^(2)=ut+(1)/(2)(2n-1)at^(2)`
Putting n=1,2,3................, we get distances travelled in 1st,2nd,3rd.........we get distances travelled in 1st,2nd,3rd....... intervals to be
`ut+(1)/(2)at^(2),ut+(5)/(2)at^(2)`.........
Thus, distance described in consecutive equal intervals of time are in A.P. with common difference `at^(2)`