8. Let $a_{n}=n \sqrt{5}-[n \sqrt{5}]$. Find the maximum and minimum terms in the sequence $a_{1}, a_{2}$, $\cdots, a_{2009}$, where $[x]$ denotes the greatest integer less than or equal to the real number $x$. (Wang Zhixiong)
8. Let $b_{0}=0, b_{1}=1$,
$$
b_{n}=4 b_{n-2}+b_{n-1}(n \geqslant 2) \text {. }
$$

Then $b_{n}=\frac{(2+\sqrt{5})^{n}-(2-\sqrt{5})^{n}}{2 \sqrt{5}}$.
In particular, $b_{6}=1292, b_{7}=5473$.
For any $k(k=1,2, \cdots, 5473)$, there exist unique integers $x_{k} 、 y_{k}$, such that
$$
1292 k=x_{k}+5473 y_{k}\left(1 \leqslant x_{k} \leqslant 5473\right) .
$$

Since $(1292,5473)=1$, it follows that $x_{1}, x_{2}, \cdots, x_{5473}$ is a permutation of $1,2, \cdots, 5473$, and the sequence $\left\{y_{k}\right\}$ satisfies
$$
y_{1} \leqslant y_{2} \leqslant \cdots \leqslant y_{5473}=1291 .
$$

For convenience, let $f(x)=x-[x]$. Then,
$$
\begin{array}{l}
f\left(x_{k} \sqrt{5}\right)=f\left(\left(1292 k-5473 y_{k}\right) \sqrt{5}\right) \\
=f\left(\frac{(2+\sqrt{5})^{6}-(2-\sqrt{5})^{6}}{2} k-\frac{(2+\sqrt{5})^{7}-(2-\sqrt{5})^{7}}{2} y_{k}\right) \\
=f\left(-(2-\sqrt{5})^{6} k+(2-\sqrt{5})^{7} y_{k}\right) . \\
\text { Also } 0<(2-\sqrt{5})^{6} k-(2-\sqrt{5})^{7} y_{k} \\
\leqslant 5473(2-\sqrt{5})^{6}-1291(2-\sqrt{5})^{7}<1,
\end{array}
$$

Thus, $f\left(x_{k} \sqrt{5}\right)=1-(2-\sqrt{5})^{6} k+(2-\sqrt{5})^{7} y_{k}$ is monotonically decreasing.
$$
\begin{array}{l}
\text { Also } x_{1}=1292, x_{5473}=5473, x_{5472}=4181, \\
x_{5471}=2889, x_{5470}=1597,
\end{array}
$$

Therefore, the maximum term is $a_{1292}$, and the minimum term is $a_{1597}$.