Using half-life period for determine the order of a reaction

Two separate experiments are performed by taking different initial concentrations of a reactant.

The progress of the reaction in each case is recorded by analysis. When the initial concentration is reduced to one-half, the time is noted. Let the initial concentrations in the two experiments be [A₁] and [A₂], while times for completion of half change are t₁and t₂ respectively.

Calculation of order of reaction. We know that half-life period for a first order reaction is independent of the initial concentration, [A]. We also know :