Question

在数列 $2001, 2002, 2003, \ldots$ 中，从第四项起，每一项等于前两项之和减去前一项。例如第四项为 $2001 + 2002 - 2003 = 2000$ 。这个数列的第 $2004$ 项是多少？

In the sequence $2001, 2002, 2003, \ldots,$ each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000.$ What is the $2004$ th term in this sequence?

$-2004$

$-2$

$0$

$4003$

$6007$

答案：C

知识点：递推等差数列找规律

难度评级：1500

解答：

递推给出 $2001$ ，、 $2002$ ，、 $2003$ ，、 $2000$ ，、 $2005$ ，、 $1998, \ldots$ 。偶数项为 $2002, 2000, 1998, \ldots$ ，每次减少 $2$ 。

第 $a_{2k} = 2004 - 2k$ 项是这些偶数项中的第 $a_{2k+1} = 2001 + 2k$ 项： $\begin{aligned} a_{2004} &= a_{2\cdot1002} \\ &= 2004 - 2(1002) = 0. \end{aligned}$

因此正确答案是 C。

The rule gives $2001,$ $2002,$ $2003,$ $2000,$ $2005,$ $1998, \ldots$ The even-indexed terms are $2002, 2000, 1998, \ldots,$ decreasing by $2.$

More precisely, the recurrence verifies inductively that $a_{2k} = 2004 - 2k$ and $a_{2k+1} = 2001 + 2k.$ Therefore $\begin{aligned} a_{2004} &= a_{2\cdot1002} \\ &= 2004 - 2(1002) = 0. \end{aligned}$

Thus, the correct answer is C.

2004 AMC 12B 第 12 题

答案：C

难度评级：1500

解答：

其他年份的第 12 题