A radioactive sample has a decay constant λ . The half-life and...

The radioactive decay law is given by $N = N_0 e^{-\lambda t}$

For half-life $T_{1/2}$, the number of active nuclei becomes half of its initial value, so $N = \dfrac{N_0}{2}$

$\dfrac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$

$e^{\lambda T_{1/2}} = 2$

$T_{1/2} = \dfrac{\ln 2}{\lambda}$

The average life or mean life $T_{avg}$ is defined as the sum of the lifetimes of all individual nuclei divided by the total number of nuclei, which is the reciprocal of the decay constant.

$T_{avg} = \dfrac{1}{\lambda}$

Thus, the half-life and average-life are $\dfrac{\ln 2}{\lambda}$ and $\dfrac{1}{\lambda}$ respectively.