Generalize Problem 20 to any number of stages.

A radioactive decay problem where the Radium decays to Radon which decays to Polonium.

The amount of polonium at time t is,

$N_3={\lambda }_1{\lambda }_2{N}_0\left[\frac{e^{-{\lambda }_{1}t}}{\left({\lambda }_2-{\lambda }_1\right)\left({\lambda }_3-{\lambda }_1\right)}+\frac{e^{-{\lambda }_{2}t}}{\left({\lambda }_1-{\lambda }_2\right)\left({\lambda }_3-{\lambda }_2\right)}+\frac{e^{-{\lambda }_{3}t}}{\left({\lambda }_1-{\lambda }_3\right)\left({\lambda }_2-{\lambda }_3\right)}\right]$

The formula for the amount of polonium at time t is given by,

${\mathit{N}}_{\mathbf{3}}\mathbf{=}{\mathit{\lambda }}_{\mathbf{1}}{\mathit{\lambda }}_{\mathbf{2}}{\mathit{N}}_{\mathbf{0}}\left[\frac{e^{-{\lambda }_{1}t}}{\left({\lambda }_2-{\lambda }_1\right)\left({\lambda }_3-{\lambda }_1\right)}+\frac{e^{-{\lambda }_{2}t}}{\left({\lambda }_1-{\lambda }_2\right)\left({\lambda }_3-{\lambda }_2\right)}+\frac{e^{-{\lambda }_{3}t}}{\left({\lambda }_1-{\lambda }_3\right)\left({\lambda }_2-{\lambda }_3\right)}\right]$

Consider the decay of Polonium. The value of the expression,

$N_3={\lambda }_1{\lambda }_2{N}_0\left[\frac{e^{-{\lambda }_{1}t}}{\left({\lambda }_2-{\lambda }_1\right)\left({\lambda }_3-{\lambda }_1\right)}+\frac{e^{-{\lambda }_{2}t}}{\left({\lambda }_1-{\lambda }_2\right)\left({\lambda }_3-{\lambda }_2\right)}+\frac{e^{-{\lambda }_{3}t}}{\left({\lambda }_1-{\lambda }_3\right)\left({\lambda }_2-{\lambda }_3\right)}\right]$ ……(1)

Replace the values,

$$\begin{gathered} {\alpha }_1=\frac{{\lambda }_1{\lambda }_2{N}_0}{\left({\lambda }_2-{\lambda }_1\right)\left({\lambda }_3-{\lambda }_1\right)} \\ {\alpha }_2=\frac{{\lambda }_1{\lambda }_2{N}_0}{\left({\lambda }_1-{\lambda }_2\right)\left({\lambda }_3-{\lambda }_2\right)} \\ {\alpha }_3=\frac{{\lambda }_1{\lambda }_2{N}_0}{\left({\lambda }_1-{\lambda }_3\right)\left({\lambda }_2-{\lambda }_3\right)} \end{gathered}$$

in equation (1), to get,

$N_3={\alpha }_1{e}^{-{\lambda }_{1}t}+{\alpha }_2{e}^{-{\lambda }_{2}t}+{\alpha }_3{e}^{-{\lambda }_{3}t}$

Find a generalization of the expressions involved. That is, we define

${\alpha }_k=\frac{{\lambda }_1{\lambda }_2\cdots {\lambda }_{n-1}{N}_0}{\left({\lambda }_{\ell }-{\lambda }_k\right)\left({\lambda }_{\ell +1}-{\lambda }_k\right)\cdots \left({\lambda }_n-{\lambda }_k\right)},n,k,\ell \in \mathrm{\mathbb{Z}}$ and denominator not equal to zero.

Apply this to generalize the expression to any number of decay stages, to get,$N_n={\alpha }_{kt}{e}^{-{\lambda }_{k}t}$