The particle starts sliding, so initial velocity=0
first, find distance travelled in n seconds,
here, u=0, t=n.
then find distance travelled in n-1 seconds.
the difference between those two is Sn.
Sn+1 denotes the distance that is travelled by particle from time n seconds to n+1 seconds.
as you have found Sn, find Sn+1.
then you get the ratio.
if you are still facing problems in understanding, ask where you need the details.
happy to help! :)

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Use the equations of motion:  s = ut + 1/2 a t^2

let Ф be the angle of the inclined plane with the horizontal.
Acceleration along the inclined plane = g sin Ф

Let initial velocity at t = 0 sec, be u = 0 m/s.
  distance travelled up to n-1 sec:  D(n-1) = 1/2 g SinФ * (n-1)²
   distance travelled up to  n sec. :  D(n) = 1/2 * g Sin Ф * n²
   Distance travelled upto n+1 sec:  D(n+1) = 1/2 * g Sin Ф * (n+1)²

 Sn = Distance travelled from t = n -1 sec. to  n sec. = 1/2 g Sin Ф * [n² - (n-1)²]
       = 1/2 g Sin Ф * (2n - 1)
Sn+1  = distance travelled from t = n sec to n+1 sec = 1/2 g Sin Ф * (2n +1)

Ratio :  Sn / Sn+1  =  (2n -1) / (2n +1)  = 1 - 1 / (n +1/2)

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