AMC12 2005 B

AMC12 2005 B · Q10

AMC12 2005 B · Q10. It mainly tests Digit properties (sum of digits, divisibility tests), 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 ${2005}^{\text{th}}$ term of the sequence?

一个数列的第一项是 $2005$。其后每一项等于前一项各位数字的立方和。求该数列的第 ${2005}^{\text{th}}$ 项。

(A) 29 29

(B) 55 55

(D) 133 133

(E) 250 250

Answer

Correct choice: (E)

正确答案：（E）

Solution

Performing this operation several times yields the results of $133$ for the second term, $55$ for the third term, and $250$ for the fourth term. The sum of the cubes of the digits of $250$ equal $133$, a complete cycle. The cycle is, excluding the first term, the $2^{\text{nd}}$, $3^{\text{rd}}$, and $4^{\text{th}}$ terms will equal $133$, $55$, and $250$, following the fourth term. Any term number that is equivalent to $1\ (\text{mod}\ 3)$ will produce a result of $250$. It just so happens that $2005\equiv 1\ (\text{mod}\ 3)$, which leads us to the answer of $\boxed{\textbf{(E) } 250}$.

多次进行该运算可得：第二项为 $133$，第三项为 $55$，第四项为 $250$。而 $250$ 的各位数字立方和等于 $133$，形成一个完整循环。除去第一项后，从第 $2^{\text{nd}}$、第 $3^{\text{rd}}$、第 $4^{\text{th}}$ 项开始，数列将以 $133$、$55$、$250$ 的顺序循环。项数与 $1\ (\text{mod}\ 3)$ 同余时对应的值为 $250$。恰好 $2005\equiv 1\ (\text{mod}\ 3)$，因此答案为 $\boxed{\textbf{(E) } 250}$。

Topics