Damon Cowan

2022-09-14

Prove $\frac{n-1}{n}+\frac{n-1}{n}\frac{n-3}{n-2}+\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}+...=\frac{n-1}{3}$ without induction
I found this identity using Maple. Is there a (simple) way to prove it without using induction? Using induction, the proof is quite easy.
Prove for odd n that
$\sum _{k=1}^{\left(n+1\right)/2}\prod _{j=0}^{k-1}\left(\frac{n-2j-1}{n-2j}\right)=\frac{n-1}{n}+\frac{n-1}{n}\frac{n-3}{n-2}+\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}+...=\frac{n-1}{3}$

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Skye Hamilton

Expert

We need to prove that
$\frac{n-1}{n}+\frac{n-1}{n}\frac{n-3}{n-2}+\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}+...=\frac{n-1}{3}$
or
$1+\frac{n-3}{n-2}+\frac{n-3}{n-2}\frac{n-5}{n-4}+...=\frac{n}{3}$
or
$\frac{n-3}{n-2}+\frac{n-3}{n-2}\frac{n-5}{n-4}+...=\frac{n-3}{3}$
or
$1+\frac{n-5}{n-4}+\frac{n-5}{n-4}\frac{n-7}{n-6}...=\frac{n-2}{3}$
or
$1+\frac{2}{3}=\frac{5}{3}.$
Done!

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