Kinematic equation for a free falling body under gravity is given by

#s=ut+1/2g t^2#

Let #s_1# be the distance traveled in time #t_1#

similarly #s_2 and s_3# be distances traveled in time #t_2 and t_3# respectively. As per original conditions

#s_1=1/2g t_1^2#

#s_2=1/2g t_2^2#

#s_3=1/2g t_3^2#

Distance traveled in time interval #t_2-t_1# #=s_2-s_1#

#=>s_2-s_1=1/2g t_2^2-1/2g t_1^2#

#=>s_2-s_1=1/2g( t_2^2- t_1^2)# ......(1)

Similarly #s_3-s_2=1/2g( t_3^2- t_2^2)# ......(2)

Ratio of the distances traveled in successive intervals can be found by dividing (1) by (2)

#(s_2-s_1)/(s_3-s_2)=(1/2g( t_2^2- t_1^2))/(1/2g( t_3^2- t_2^2))#

#(s_2-s_1)/(s_3-s_2)=( t_2^2- t_1^2)/( t_3^2- t_2^2)#