AMC 8 1998 Problem 23
AMC8 1998 年第 23 题
数列与规律★★★☆☆
If the pattern in the diagram continues, what fraction of eighth triangle would be shaded?
如果图中的图案继续下去，第八个三角形着色部分的比例是多少？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
In each phase, all small triangles are congruent. The number of shaded triangles follows the pattern:
0, 1, 3, 6,...
which is the pattern of "triangular numbers". Each time, the number 1, 2, 3, 4, 5... is added to the previous term. Thus, the first eight terms are:
0, 1, 3, 6, 10, 15, 21, 28
In the eighth diagram, there will be 28 shaded triangles.
The total number of small triangles follows the pattern:
1, 4, 9, 16,...
which is the pattern of "square numbers". Thus, the eighth triangle will be divided into small triangles in total.
The ratio of shaded to total triangles will be the fraction of the whole figure that's shaded, since all triangles are congruent. Thus, the answer is = , and the correct choice is
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