General term of a sequence $(2-1)(2+1)(3-1)(3+1)...(n-1)(n+1)$

Question

Can we use integrals, and are there some general methods for finding terms of a sequence?

Answer 1

Let's reorder the terms and recognize the result:

$$ \begin{align}P&=(2-1)(2+1)(3-1)(3+1)\dots(n-1)(n+1) \\&=(2-1)(3-1)\dots(n-1)\times(2+1)(3+1)\dots(n+1) \\&=\frac{(n-1)!(n+1)!}{2} \end{align} $$