Answer
$a_n$ converges to $0$.
Work Step by Step
We have
$$
\lim _{n \rightarrow \infty} a_n=\lim _{n \rightarrow \infty}
\frac{100^{n}}{n !}-\frac{3+\pi^{n}}{5^{n}}
\\
=\lim _{n \rightarrow \infty}
\frac{100^{n}}{n !}-\lim _{n \rightarrow \infty} \frac{3}{ 5^{n}}- \lim _{n \rightarrow \infty} (\frac{\pi}{5})^{n}
=0.
$$
Where we used the facts: for large $n\gt \gt 100$ , $n!$ is getting bigger faster than $100^n$; also, $\pi/5\lt 1$.
Thus, $a_n$ converges to $0$.
