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! :)