Find n(Ni) in Decay Chain with Single Equation

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The discussion focuses on calculating the number of particles of Ni, denoted as n(Ni), in a decay chain represented by the equation N1 -> N2 -> N3 -> ... -> Ni -> ... -> Nd. The Bateman equations are referenced as a potential solution, but they fail when i equals d due to division by zero as lambda approaches zero. A viable alternative involves introducing an additional decay to Nd+1 and considering the limit of an infinite lifetime for Nd, or alternatively, subtracting the contributions of all other elements from the total particle count.

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For the decay chain N1 -> N2 -> N3 -> ... -> Ni -> ... -> Nd, how can I find the amount of particles of Ni, n(Ni), at any point in time t, with a single equation where i can vary from 1 to d? I have already seen WIkipedia's suggestion on the Bateman equations but that method seems to collapse for the case when i=d (as lambda=0 so we'd be dividing by 0).
 
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You can use the equation with an additional decay to Nd+1, and take the limit of an infinite lifetime of Nd. Alternatively, subtract all the other elements from the total number of particles.
 

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