A particle starts from rest and moves with uniform acceleration.  Then the ratio of distance covered in n^th second to n seconds is 

Using the equations of  motion

Distance travelled by a body in n^th second is 

${\mathrm{s}}_{\mathrm{n}}=\mathrm{u}+\frac{\mathrm{a}}{2}(2\mathrm{n}-1)$

Distance travelled in 'n' seconds is

$\mathrm{s}=\mathrm{un}+\frac{1}{2}{\mathrm{an}}^2$

u is zero since the body starts from rest.

$\frac{{\mathrm{s}}_{\mathrm{n}}}{\mathrm{s}}=\frac{\frac{\mathrm{a}}{2}(2\mathrm{n}-1)}{\frac{1}{2}{\mathrm{an}}^2}=\frac{2}{\mathrm{n}}-\frac{1}{{\mathrm{n}}^2}$