AMC10 2005 B
AMC10 2005 B · Q11
AMC10 2005 B · Q11. It mainly tests Perfect squares & cubes, Sequences in number theory (remainders patterns).
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 2005th term of the sequence?
一个数列的第一项是 2005。每项之后的项是前一项各位数字立方的和。这个数列的第 2005 项是多少？
(A)
29
29
(B)
55
55
(C)
85
85
(D)
133
133
(E)
250
250
Answer
Correct choice: (E)
正确答案：（E）
Solution
The sequence begins 2005, 133, 55, 250, 133, ... Thus after the initial term 2005, the sequence repeats the cycle 133, 55, 250. Because $2005 = 1 + 3 \cdot 668$, the 2005th term is the same as the last term of the repeating cycle, 250.
数列开始是 2005, 133, 55, 250, 133, ... 因此在初始项 2005 之后，数列重复循环 133, 55, 250。因为 $2005 = 1 + 3 \cdot 668$，第 2005 项与重复循环的最后一项相同，即 250。
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