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

小提示：

先计算前几项，直到数值开始重复。

Compute the first several terms until the values start repeating

大提示：

找出循环后，将第一项之后的步数除以循环长度并取余。

Once you find the repeating cycle, reduce the number of steps after the first term modulo its length

解答：

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

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

小提示：

大提示：

解答：

其他年份的第 11 题