Recently, I've been trying to reason about different factorial values.
For example, intuitively, it seems to me that:
$$(n-1)! < \frac{\left(n+\frac{2n}{3}\right)!}{n!}$$
What would be the right approach to demonstrate this?
If I go with induction, the base case is straight forward. This is clearly true for $n=\{3,4,5\}$
For the inductive step, I can prove it holds if $n$ is of form $3i$ or form $3i+2$. I am not able to show it for the case where $n$ is of the form $3i+1$.
If my intuition is correct, what is the easiest way to complete the argument? Would the gamma function, for example, help me here?
If my intuition is not correct, could you show me how to analyze the conditions where it does not hold.