19.2.5 * Given that $n$ is a positive integer. Let $A_{n}=(7+4 \sqrt{3})^{n}$. Simplify $1+\left[A_{n}\right]-A_{n}$.
Parse $(7+4 \sqrt{3})^{n}+(7-4 \sqrt{3})^{n}$ is an integer, and $1>7-4 \sqrt{3}>0$. Therefore,
$$
\begin{array}{c}
{\left[(7+4 \sqrt{3})^{n}\right]=(7+4 \sqrt{3})^{n}+(7-4 \sqrt{3})^{n}-1} \\
1+\left[A_{n}\right]-A_{n}=(7-4 \sqrt{3})^{n} .
\end{array}
$$