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A radioactive nucleus can decay by two different processes. The half lives of the first and second decay processes are $5 \times 10^3$ and $10^5$ years respectively. Then, the effective half-life of the nucleus is
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$4762 \mathrm{yr}$
The effective half-life of the nucleus
$\begin{aligned}
\lambda & =\lambda_1+\lambda_2 \\
\frac{0.69}{T} & =\frac{0.69}{T_1}+\frac{0.69}{T_2} \\
T & =\frac{T_1 T_2}{T_1+T_2} \\
& =\frac{5 \times 10^3 \times 10^5}{5 \times 10^3[1+20]}=\frac{5 \times 10^5}{105} \\
& =4762 \mathrm{yr}
\end{aligned}$
$\begin{aligned}
\lambda & =\lambda_1+\lambda_2 \\
\frac{0.69}{T} & =\frac{0.69}{T_1}+\frac{0.69}{T_2} \\
T & =\frac{T_1 T_2}{T_1+T_2} \\
& =\frac{5 \times 10^3 \times 10^5}{5 \times 10^3[1+20]}=\frac{5 \times 10^5}{105} \\
& =4762 \mathrm{yr}
\end{aligned}$
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