AMC10 2001 A

AMC10 2001 A · Q11

AMC10 2001 A · Q11. It mainly tests Arithmetic sequences basics, Basic counting (rules of product/sum).

Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 8 unit squares. The second ring contains 16 unit squares. If we continue this process, the number of unit squares in the $100^{th}$ ring is

考虑一个单位正方形阵列中的深色正方形，部分显示如下。围绕这个中心正方形的第一个环包含 8 个单位正方形。第二个环包含 16 个单位正方形。如果我们继续这个过程，第 $100^{th}$ 环中的单位正方形数量是

(A) 396 396

(B) 404 404

(D) 10,000 10,000

(E) 10,404 10,404

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) The $n^{th}$ ring can be partitioned into four rectangles: two containing $2n+1$ unit squares and two containing $2n-1$ unit squares. So there are $$2(2n+1)+2(2n-1)=8n$$ unit squares in the $n^{th}$ ring. Thus, the $100^{th}$ ring has $8\cdot 100=800$ unit squares.

（C）第 $n^{th}$ 圈可以分成四个长方形：两个包含 $2n+1$ 个单位方格，两个包含 $2n-1$ 个单位方格。因此共有 $$2(2n+1)+2(2n-1)=8n$$ 个单位方格在第 $n^{th}$ 圈中。因此，第 $100^{th}$ 圈有 $8\cdot 100=800$ 个单位方格。

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