2005 AMC 10B 第 11 题

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

先计算前几项，直到数值开始重复。

Compute the first several terms until the values start repeating

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

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.

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