Question

How do you simplify the factorial expression `((2n-1)!)/((2n+1)!)`?

Answer 1

`((2n-1)!)/((2n+1)!) = 1/((2n+1)(2n))`

Explanation:

Remember that:

`n! =n(n-1)(n-2)...1`

And so

`(2n+1)! =(2n+1)(2n)(2n-1)(2n-2) ... 1`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=(2n+1)(2n)(2n-1)!`

So we can write:

`((2n-1)!)/((2n+1)!) = ((2n-1)!)/((2n+1)(2n)(2n-1)!)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =1/((2n+1)(2n))`

Example: `n=10`

`LHS = (19!)/(21!) \ \ \ \ \ = 1/420`
`RHS = 1/(21*20) = 1/420`