Measurements are reported for the first time for the D.C. Kerr response and the pretransitional behaviour of a new homologous series of liquid crystal 4'-(trans-4'-alkylcyclohexyl)-benzonitrile(for n=5 to 9) at and above the nematic-isotropic phase transition temperatures. The real part of the third order nonlinear susceptibilities(chi ((3))) have been determined for this homologous series of liquid crystals from the Electro-optic Kerr effect (EOKE) experiments at and above the nematic-isotropic phase transition temperatures. The variation of the Kerr constant with temperature has been discussed on the basis of the Landau-de Gennes model of nematic -isotropic phase transition. The Kerr relaxation time and other statistical parameters concerned with a nematic-isotropic phase transition like the latent heat of transition have also been determined using the Landau-de Gennes' theory of phase transition. The variation of the Kerr constant, the values of the third order nonlinear susceptibilities, the Kerr relaxation time and the latent heat of transition with the number of carbon atoms in the alkyl chain within the series have been discussed on the basis of an interesting effect called the "ODD-EVEN" effect. The results and discussions are presented.