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Re: induction on finite set.
Posted:
Feb 26, 2013 5:21 AM


The principle of finite induction can be derived from the fact that every nonempty set of natural numbers has a smallest element. This fact is known as the wellordering principle for natural numbers. The finite induction therefore is not that much brader. For any given positive integer N, we have 2N1 real variables x_k \in [0, 1], with 1 \leq k \leq 2N1. We also know x_1 = 1 = x_{2N1}, and for all other k, x_k < 1. Additionally, we may use the notation S_k = \sum_{j = 1}^{k} x_j We have a recurrence that is valid for 1 \leq k \leq 2N  2: S_k = x_k + (1  x_k)S_{k+1}
We can manipulate that in a variety of ways, such as the following which are valid (if I haven?t made a mistake) when 1 < k < 2N  1 If our aim is to find a general form of x_k as a function of k, for each positive integer N. A general form for S_k would be nice as well. My preliminary attempts suggest perhaps there will be symmetry, with x_{Nn} = x_{N+n}, and I would like to prove or disprove that. I also suspect that there will be N1 polynomials p_k of degree N, such that for 1 < k \leq N we will have p_k(x_k) = 0. I?d like to know whether that is true or false.
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