Question

一个数列的第一项为 $2005$ 。之后每一项都等于前一项各位数字的立方和。这个数列的第 $2005$ 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：递推数字找规律

难度评级：1370

解答：

数列开头是 $2005, 133, 55, 250, 133, \ldots,$ 所以从第一项以后，它重复长度为 $3.$ 的循环 $133, 55, 250$ 。

第 $2, 3,$ 和 $4$ 项分别是这个循环的第一、第二和第三项。因为 $2005 - 2 = 2003$ 除以 $3,$ 的余数是 $2$ ，所以第 $2005$ 项对应循环的第三项，即 $250.$

所以正确答案是 E。

The sequence begins $2005, 133, 55, 250, 133, \ldots,$ so after the first term it repeats the cycle $133, 55, 250$ of length $3.$

Terms $2, 3,$ and $4$ are the first, second, and third entries of this cycle. Because $2005 - 2 = 2003$ leaves remainder $2$ upon division by $3,$ the $2005$ th term matches the third entry, $250.$

Thus, E is the correct answer.

2005 AMC 10B 第 11 题

答案：E

难度评级：1370

解答：

其他年份的第 11 题