AMC 8 1985 Problem 7
AMC8 1985 年第 7 题
数列与规律★★☆☆☆
A "stair-step" figure is made of alternating black and white squares in each row. Rows 1 through 4 are shown. All rows begin and end with a white square. The number of black squares in the row is
一个"阶梯"图形由每行交替的黑白正方形组成。图中显示了第 1 行到第 4 行。每行都以一个白色正方形开始和结束。第 37 行中黑色正方形的数量是
- A.34
- B.35
- C.36
- D.37
- E.38
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
The first row has 0 black squares, while each row after the first has 1 more black square than the previous row. Therefore, the second row has 1 black square, the third row has 2 black squares, and so on, until the row has 36 black squares.
Let be the number of black squares in the row. Since each row has 1 more black square than the previous row, is linear with slope 1, that is, f(n) = n + b for some constant . Since f(1) = 0, 0 = 1 + b and b = -1. So f(n) = n-1 and the row has f(37) = 37 - 1 = 36 black squares.
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