Answer
The sequence $a_n$ is monotonic and unbounded.
Work Step by Step
Consider $a_n=\dfrac{(2n+3)!}{(n+1)!}$ for all $n \in N$
Also, $a_{n+1}=\dfrac{(2(n+1)+3)!}{((n+1)+1)!}$
and $(4n+10)\dfrac{(2n+3)!}{(n+1)!}=(4n+10) a_n$ ;
for all $n \in N$
so, $1 \lt 4n+10$ for all $n \in N$
This means that $a_n \lt a_{n+1}$ for all $n \in N$ .Thus $a_n$ increases and is monotonic.
To check the sequence is bounded or not.In order to find that we have $a_n \lt \dfrac{(2n+3)!}{(n+1)!}=(2n+3)(\dfrac{(2n+2)!}{(n+1)!})$
Also, $\lim\limits_{n \to \infty}2n+3=\infty$
so, $a_n$ is unbounded.
Therefore, we can see that the sequence $a_n$ is monotonic and unbounded.
