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How do I go from the following recurrence relation

a(n) = (n+1)a(n-1) where a(0) = 2

to a closed form? I know I need to use an iterative approach but I am not quite sure what to look for.

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We can divide through by the summation factor $(n+1)!$ to get $${a_n\over (n+1)!}=\frac{a_{n-1}}{n!}$$ Now let $$S_n:=\frac{a_n}{(n+1)!}$$ Thus $S_n=S_{n-1}=c$ is constant, but $a_0=2$ and so $S_0=2/1=2\implies S_n\equiv2$ Thus we have $$a_n=(n+1)!S_n=2\cdot (n+1)!$$

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  • $\begingroup$ Thank you, and if I wanted to prove that this answer was correct, would I do that with a proof by induction or what? $\endgroup$ – DrJonesYu Apr 28 '14 at 1:20
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    $\begingroup$ This is sufficient proof in my opinion. You could certainly verify it that way, but there's nothing wrong with a direct derivation. $\endgroup$ – user142299 Apr 28 '14 at 1:24
  • $\begingroup$ What is the approach called that you took to solve this? $\endgroup$ – DrJonesYu Apr 28 '14 at 1:31
  • $\begingroup$ I'm not sure - I learned it originally from the book "Concrete Mathematics" by Knuth, et al. See this: homepages.gac.edu/~holte/courses/mcs256/documents/… They show how to find $S_n$ in general. This method can be used to solve any recurrence of the form $a_nx_n=b_nx_{n-1}+c_n$ so it's fairly powerful and nice to know. $\endgroup$ – user142299 Apr 28 '14 at 1:34

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