1. Find the value of $\frac{1^{4}+2009^{4}+2010^{4}}{1^{2}+2009^{2}+2010^{2}}$.
(1 mark)
求 $\frac{1^{4}+2009^{4}+2010^{4}}{1^{2}+2009^{2}+2010^{2}}$ 的值。
(1 分)
1. 4038091
1. We have
$$
\begin{aligned}
\frac{1^{4}+a^{4}+(a+1)^{4}}{1^{2}+a^{2}+(a+1)^{2}} & =\frac{1^{4}+a^{4}+a^{4}+4 a^{3}+6 a^{2}+4 a+1}{1^{2}+a^{2}+a^{2}+2 a+1} \\
& =\frac{2\left(a^{4}+2 a^{3}+3 a^{2}+2 a+1\right)}{2\left(a^{2}+a+1\right)} \\
& =\frac{\left(a^{2}+a+1\right)^{2}}{a^{2}+a+1} \\
& =a^{2}+a+1
\end{aligned}
$$
and so setting $a=2009$ gives the answer $2009^{2}+2009+1=4038091$.
Remark. It would be easier to start by plugging in small values of $a$ and then try to look for a pattern.