At time t = 0, N₁ nuclei of decay constant ${\lambda }_1 \mathrm{\&} {\mathrm{N}}_2$ nuclei of decay constant ${\lambda }_2$ are mixed. The decay constant of the mixture is 

At t = 0; Number of nuclei = N₁ 
After a time ‘t’, Number of remaining nuclei 
$\text{ is }{\mathrm{N}}_1^1={\mathrm{N}}_1\left({\mathrm{e}}^{-{\lambda }_1\mathrm{t}}\right)$
$\text{ In case of }{\mathrm{N}}_2\text{ nuclei, afte|r }{\mathrm{N}}_2^1={\mathrm{N}}_2\left({\mathrm{e}}^{-{\lambda }_2\mathrm{t}}\right)$
Total decay rate after mixing the two nuclei 
$\text{ after a time }\mathrm{t}\text{ is }\lambda \mathrm{N}={\lambda }_1{\mathrm{N}}_1^1+{\lambda }_2{\mathrm{N}}_2^1$
$\mathrm{\lambda }\left({\mathrm{N}}_1^1+{\mathrm{N}}_2^1\right)={\mathrm{\lambda }}_1\left[{\mathrm{N}}_1{\mathrm{e}}^{-{\mathrm{\lambda }}_1\mathrm{t}}\right]+{\mathrm{\lambda }}_2\left[{\mathrm{N}}_2{\mathrm{e}}^{-{\mathrm{\lambda }}_2\mathrm{t}}\right]$
$\lambda =\frac{{\lambda }_1{N}_1{e}^{-{\lambda }_{1}t}+{\lambda }_2{N}_2{e}^{-{\lambda }_{2}t}}{N_1{e}^{-{\lambda }_{1}t}+N_2{e}^{-{\lambda }_{2}t}}=\frac{N_1{\lambda }_1{e}^{-{\lambda }_{1}t}+N_2{\lambda }_2{e}^{-{\lambda }_{2}t}}{N_1{e}^{-{\lambda }_{1}t}+N_2{e}^{-{\lambda }_{2}t}}$