Question

一个数列的第一项是 $2005$ 。每一项之后的下一项等于前一项各位数字的立方和。该数列的第 $2005$ 项是多少？

The first term of a sequence is $2005.$ Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2005$ th term of the sequence?

$29$

$55$

$85$

$133$

$250$

答案：E

知识点：数字递推找规律

难度评级：1440

解答：

数列开始为 $2005, 133, 55, 250, 133, \ldots$ ，因为 $2^3 + 0^3 + 0^3 + 5^3 = 133$ ， $1^3 + 3^3 + 3^3 = 55$ ， $5^3 + 5^3 = 250$ ，且 $2^3 + 5^3 + 0^3 = 133$ 。

在首项 $2005$ 之后，数列以 $133, 55, 250$ 为周期 $3$ 循环。

当 $n \ge 2$ 时，第 $n$ 项是序列 $133, 55, 250$ 中索引为 $((n-2)\bmod 3)$ 的一项。因为 $2005 - 2 = 2003 \equiv 2 \pmod 3$ ，所以第 $2005$ 项是 $250$ 。

所以正确答案是 E。

The sequence begins $2005, 133, 55, 250, 133, \ldots$ since $2^3 + 0^3 + 0^3 + 5^3 = 133,$ $1^3 + 3^3 + 3^3 = 55,$ $5^3 + 5^3 = 250,$ and $2^3 + 5^3 + 0^3 = 133.$

After the initial $2005,$ the terms cycle through $133, 55, 250$ with period $3.$

Term $n$ for $n \ge 2$ is the $((n-2)\bmod 3)$ th entry of $133, 55, 250.$ Since $2005 - 2 = 2003 \equiv 2 \pmod 3,$ the $2005$ th term is $250.$

Thus, the correct answer is E.

2005 AMC 12B 第 10 题

答案：E

难度评级：1440

解答：

其他年份的第 10 题