Question

图形 $0, 1, 2$ 和 $3$ 分别由 $1, 5, 13$ 和 $25$ 个互不重叠的单位正方形组成。如果继续这个规律，图形 $100$ 中会有多少个互不重叠的单位正方形？

Figures $0, 1, 2,$ and $3$ consist of $1, 5, 13,$ and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure $100?$

$10401$

$19801$

$20201$

$39801$

$40801$

答案：C

知识点：前n个奇数之和完全平方数找规律

难度评级：1240

解答：

图形 $n$ 可以看作前 $n$ 个奇数之和加上前 $n+1$ 个奇数之和，所以单位正方形个数为 $n^2 + (n+1)^2$ 。

对图形 $100$ 有 $\begin{aligned} 100^2 + 101^2 &= 10000 + 10201 \\ &= 20201. \end{aligned}$

所以正确答案是 C。

Figure $n$ can be split into the sum of the first $n$ odd numbers and the first $n+1$ odd numbers, giving $n^2 + (n+1)^2$ unit squares.

For figure $100,$ this is $\begin{aligned} 100^2 + 101^2 &= 10000 + 10201 \\ &= 20201. \end{aligned}$

Thus, the correct answer is C.

2000 AMC 10 第 12 题

答案：C

难度评级：1240

解答：

其他年份的第 12 题