Answer
The sequence $a_n$ is monotonic and unbounded.
Work Step by Step
Our aim is to find out if the sequence $a_n$ is monotonic.
In order to this, we will consider $a_n=\dfrac{(2n+3)!}{(n+1)!}$ for all $n \in N$
Also, $a_{n+1}=\dfrac{(2(n+1)+3)!}{((n+1)+1)!}=(4n+10)\dfrac{(2n+3)!}{(n+1)!}=(4n+10) a_n$
for all $n \in N$
Thus, we find that $1 \lt 4n+10$ for all $n \in N$
This implies that $a_n \lt a_{n+1}$ for all $n \in N$ and so, $a_n$ is increasing and thus, monotonic.
Now, we will check for a bounded or unbounded sequence. In order to this we will take $a_n \lt \dfrac{(2n+3)!}{(n+1)!}=(2n+3)(\dfrac{(2n+2)!}{(n+1)!})$ and $\lim\limits_{n \to \infty}2n+3=\infty$
Thus, $a_n$ is unbounded.
Hence, the sequence $a_n$ is monotonic and unbounded.
