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

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

解答：

其他年份的第 3 题