Example 2-37 Find the value of the following $n$-order determinant:
$$
d_{n}=\left|\begin{array}{llllll}
2 & 1 & 0 & 0 & \cdots & 0 \\
1 & 2 & 1 & 0 & \cdots & 0 \\
0 & 1 & 2 & 1 & \cdots & 0 \\
& & & \vdots & & \\
0 & 0 & 0 & 0 & \cdots & 2
\end{array}\right|
$$

According to the properties of determinants