In a radioactive material the activity at time t1 is R1 and at a later time t2, it is R2. If the decay constant of the material is λ,then

The decay rate R of a radioactive material is the number of decays per second.From radioactive decay law,-dNdt∝N or -dNdt=λN Thus,R=- dNdt or R∝N or R=λN or R=λN0e-λt    .....(i)Where R0=λN0 is the activity of the radioactive material at time t = 0.At time t1, R1=R0e-λt1    ......(ii)At time t2, R2=R0e-λt2      ......(iii)Dividing Eq.(ii) by (iii) , we have R1R2=e-λt1e-λt2=e-λ(t1-t2) or R1=R2e-λ(t1-t2)