A particle starts sliding down a frictionless inclined plane. If $${S_n}$$  is the distance travelled by it from time $$t = n - 1\,sec$$    to $$t = n\,sec,$$   the ratio $$\frac{{{S_n}}}{{{S_{n + 1}}}}$$  is-

Solution :
$$\eqalign{ & {s_n} = \frac{a}{2}\left( {2n - 1} \right); \cr & {s_{n\, + \,1}} = \frac{a}{2}\left[ {2\left( {n + 1} \right) - 1} \right] = \frac{a}{2}\left( {2n + 1} \right) \cr & \frac{{{s_n}}}{{{s_{n\, + \,1}}}} = \frac{{2n - 1}}{{2n + 1}} \cr} $$