A radioactive sample can decay by two different processes. The half-life for the first process is $T_1$ and that for the second process is $T_2$. The effective half-life $T$ of the radioactive sample is

Radioactive decay rate is directly proportional to number of nucleus present at any time

$\frac{dN}{dt}=-\lambda N$....(i)

where $\lambda$ is a constant related to half life as $\lambda =\frac{\ln 2}{T_{\frac{1}{2}}}$ where $T_{\frac{1}{2}}$ is half life time of radioactive decay

The total rate of decay is

$\frac{dN}{dt}=-({\lambda }_1+{\lambda }_2)N={\lambda }_{effective}N$...(ii)

${\lambda }_1=\frac{\ln 2}{T_1}$

${\lambda }_2=\frac{\ln 2}{T_2}$

${\lambda }_{effective}=\frac{\ln 2}{T_{effective}}$

substituting in equation (ii)

$(\frac{\ln 2}{T_1}$ $+\frac{\ln 2}{T_2})N$ $=\frac{\ln 2}{T_{effective}}N$

$\frac{1}{T}=\frac{1}{T_1}+\frac{1}{T_2}$