velocity of particle at time t = ft

at time 2t = ft + 2ft

at time 3t = ft + 2ft + 3ft

at time nt = ft + 2ft + 3ft  +  ...... + nt

so velocity at time nt = n(n+1)/2 * ft

distance travelled = integral zero to t (n(n+1)/2 * ft) dt = n(n+1)/4 * ft^2