Find n(Ni) in Decay Chain with Single Equation

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To find the amount of particles of Ni in a decay chain at time t, a single equation approach is needed, especially when i varies from 1 to d. The Bateman equations are commonly referenced but fail when i equals d due to division by zero. An alternative method involves introducing an additional decay to Nd+1 and considering the limit of Nd's infinite lifetime. Another approach is to calculate the total number of particles and subtract the contributions from all other elements in the decay chain. These methods provide viable solutions for determining n(Ni) at any given time.
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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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