Question

考虑正整数序列

$\begin{gathered} 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, \\ 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, \\ 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, \\ 1, 2, \ldots \end{gathered}$

这个序列的第 $2025$ 项是多少？

Consider the sequence of positive integers

$\begin{gathered} 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, \\ 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, \\ 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, \\ 1, 2, \ldots \end{gathered}$

What is the $2025$ th term in this sequence?

$5$

$15$

$16$

$44$

$45$

答案：E

知识点：找规律完全平方数

难度评级：1200

解答：

按块分组。第 $k$ 块为 $k, k-1, \ldots, 2, 1, 2, \ldots, k-1, k$ ，共有 $2k - 1$ 项，并以 $k$ 结束。累计到第 $k$ 块末尾共用了 $1 + 3 + \cdots + (2k-1) = k^2$ 项。注意 $2025 = 45^2$ 。这正好是第 $45$ 块的末尾，而该块的最后一项为 $45$ 。因此正确答案是 E。

Group the sequence into blocks. Block $k$ reads $k, k-1, \ldots, 2, 1, 2, \ldots, k-1, k,$ which is $2k - 1$ terms and ends on $k.$ So after block $k$ we've used $1 + 3 + \cdots + (2k-1) = k^2$ terms. Notice $2025 = 45^2.$ That's exactly the end of block $45,$ whose last term is $45.$ Thus, E is the correct answer.

2025 AMC 10A 第 5 题

答案：E

难度评级：1200

解答：

其他年份的第 5 题