The activity of a radioactive sample is A1 at time t1 and A2 at timet2. If τ is average life of sample then the number of nuclei decayed in time (t2−t1) is

[d] Let N0 be the initial number of nuclei, then N1=N0e−λt1 and N2=N0e−λt2 ∴  Number of nuclei decayed =N1−N2 =N0(e−λt1−e−λt2)=A0λ(e−λt1−e−λt2) =A1−A2λ=(.A1−A2)τ.