2015 AMC 10A 第 3 题

Question

Ann 如图用

18

根牙签搭了一个

3

级楼梯。她还需要再加多少根牙签才能完成一个

5

级楼梯？

Ann made a

3

-step staircase using

18

toothpicks as shown in the figure. How many toothpicks does she need to add to complete a

5

-step staircase?

9

18

20

22

24

答案：D

知识点：等差数列找规律

难度评级：960

小提示：

观察下一阶楼梯会新增多少根牙签。

Look at how many new toothpicks the next stair step adds

大提示：

每增加一级，新添的牙签数比前一次多

2

根。

The number of new toothpicks increases by

2

with each added step

解答：

我们试着找出楼梯级数与所需牙签数之间的规律。

对于

1

级楼梯，只需要

4

根牙签，也就是一个正方形。

对于

2

级楼梯，根据图形需要

10

根牙签。

同理，

3

级楼梯需要

18

根牙签。

2

级楼梯比

1

级楼梯多用

10 - 4 = 6

根牙签。

3

级楼梯比

2

级楼梯多用

18 - 10 = 8

根牙签。

按照这个规律，

4

级楼梯需要

18 + 10 = 28

根牙签，

5

级楼梯需要

28 + 12 = 40

根牙签。

这说明 Ann 还需要添加

40 - 18 = 22

根牙签。

所以正确答案是 D。

Let us try to find a pattern between the number of toothpicks needed for the staircases.

For a

1

-step staircase, we would only need

4

toothpicks (just a square).

For a

2

-step staircase, we would need

10

toothpicks according to the diagram.

Similarly, we would need

18

toothpicks for a

3

-step staircase.

A

2

-step staircase needs

10 - 4 = 6

more toothpicks than a

1

-step staircase. A

3

-step staircase needs

18 - 10 = 8

more toothpicks than a

2

-step staircase.

Following this pattern, we can see that a

4

-step staircase will need

18 + 10 = 28

toothpicks, and a

5

-step staircase will need

28 + 12 = 40

toothpicks.

This means that Ann would need to add

40 - 18 = 22

more toothpicks.

Thus, D is the correct answer.

2015 AMC 10A 第 3 题

答案：D

难度评级：960

小提示：

大提示：

解答：

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