section_path: Mathematics/Universal law/Chain-decay processes/Chain of any number of decays
section_name: Chain of any number of decays
paragraph_text: Chain of any number of decays
  
  For the general case of any number of consecutive decays in a decay chain, i.e. A → A ··· → A ··· → A, where D is the number of decays and i is a dummy index (i = 1, 2, 3,..., D), each nuclide population can be found in terms of the previous population. In this case N = 0, N = 0,..., N = 0. Using the above result in a recursive form:
  
  The general solution to the recursive problem is given by Bateman's equations:
  
  N D = N 1 (0) λ D ∑ i = 1 D λ i c i e − λ i t c i = ∏ j = 1, i ≠ j D λ j λ j − λ i {\displaystyle {\begin{aligned}N_{D}&={\frac {N_{1}(0)}{\lambda _{D}}}\sum _{i=1}^{D}\lambda _{i}c_{i}e^{-\lambda _{i}t}\\c_{i}&=\prod _{j=1,i\neq j}^{D}{\frac {\lambda _{j}}{\lambda _{j}-\lambda _{i}}}\end{aligned}}}