Example 4. If it is known that $a_{1}=3, a_{n}=3^{4 \mathrm{n}^{2}+1} a_{\mathrm{n}-1}$, find $a_{\mathrm{A}}$.


Given the conditions, we know $k=1, a_{\mathbf{k}}=a_{1}=3, b=3$.
Substituting into formula (18), we get
$$
\begin{aligned}
a_{\mathrm{n}} & =3 \cdot 3^{\sum_{\mathrm{i}=2}^{\mathrm{n}}\left(4 n^{2}+1\right)} \\
& =3 \cdot 3^{4 \cdot \frac{n}{6}(2 n+1)(n+1)+n-5} \\
& =3^{\frac{1}{3}\left(4 n^{3}+4 n^{2}+5 n-12\right) .}
\end{aligned}
$$