A particle moves from rest in a straight line with alternate acceleration and retardation of mahnitudes `f` and `f'` during equal intervals of time `t`. At the end of `2n` such intervals prove that the space it has described is
`(n t^(2))/(2) [(2n + 1) f - (2 n - 1)f']`

Explanation

`S_(1) = (1)/(2) xx f xx t^(2)`
`S_(2) = t (ft) - (1)/(2) f' t^(2)`
The distance travelled in 1 st `2t` sec
`S' = S_(1) + S_(2) = (ft^(2))/(2) - (1)/(2) f't^(2)`
Similarly, the distance travelled in next `2t` sec
`S'' = (2t^(2))/(2) [5 f - 3 f']`
Hence, total space `= (n t^(2))/(2) [(2n + 1)f - (2n - 1) f']`