A radioactive sample disintegrates via two independent decay processes having half lives $$T_{1/2}^{(1)}$$ and $$T_{1/2}^{(2)}$$ respectively. The effective half-life $$T_{1/2}$$ of the nuclei is:

When a radioactive sample decays via two independent decay processes, the total decay constant is the sum of the individual decay constants: $$\lambda = \lambda_1 + \lambda_2$$.

Since the decay constant is related to the half-life by $$\lambda = \frac{\ln 2}{T_{1/2}}$$, we have $$\frac{\ln 2}{T_{1/2}} = \frac{\ln 2}{T_{1/2}^{(1)}} + \frac{\ln 2}{T_{1/2}^{(2)}}$$. Dividing through by $$\ln 2$$: $$\frac{1}{T_{1/2}} = \frac{1}{T_{1/2}^{(1)}} + \frac{1}{T_{1/2}^{(2)}} = \frac{T_{1/2}^{(1)} + T_{1/2}^{(2)}}{T_{1/2}^{(1)} \cdot T_{1/2}^{(2)}}$$.

Therefore $$T_{1/2} = \frac{T_{1/2}^{(1)} \cdot T_{1/2}^{(2)}}{T_{1/2}^{(1)} + T_{1/2}^{(2)}}$$.