AMC12 2000 A

AMC12 2000 A · Q8

AMC12 2000 A · Q8. It mainly tests Sequences & recursion (algebra), Arithmetic misc.

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?

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。如果继续这个模式，图形 100 中会有多少个互不重叠的单位正方形？

(A) 10401 10401

(B) 19801 19801

(D) 39801 39801

(E) 40801 40801

Answer

Correct choice: (C)

正确答案：（C）

Solution

We can attempt $0^2+1^2=1$ and $1^2+2^2=5$, so the pattern here looks like the number of squares in the $n$-th figure is $n^2+(n+1)^2$. When we plug in 100 for $n$, we get $100^2+101^2=10000+10201=20201$, or option $\textbf{(C)}$.

尝试 $0^2+1^2=1$ 且 $1^2+2^2=5$，因此这里的规律看起来是第 $n$ 个图形中的正方形数量为 $n^2+(n+1)^2$。将 100 代入 $n$，得到 $100^2+101^2=10000+10201=20201$，对应选项 $\textbf{(C)}$。

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