Question

Sara 用牙签搭了如下楼梯：

这是一个 $3$ 级楼梯，使用了 $18$ 根牙签。如果一个楼梯使用了 $180$ 根牙签，它有多少级？

Sara makes a staircase out of toothpicks as shown:

This is a $3$ -step staircase and uses $18$ toothpicks. How many steps would be in a staircase that used $180$ toothpicks?

$10$

$11$

$12$

$24$

$30$

答案：C

知识点：二次方程求和找规律

难度评级：1200

解答：

$n$ 级楼梯中，竖直牙签数为 $(1 + 2 + \cdots + n) + n$ $= \tfrac{n(n+1)}{2} + n$ ，水平牙签数与其相同，因此总数为 $n(n+1) + 2n = n(n+3)$ 。检验可知 $n = 3$ 时有 $18$ 根牙签。令 $n(n+3) = 180$ 。因式分解得 $(n-12)(n+15) = 0$ ，所以 $n = 12$ 。正确答案是 C。

In an $n$ -step staircase the vertical toothpicks number $(1 + 2 + \cdots + n) + n$ $= \tfrac{n(n+1)}{2} + n,$ and there are just as many horizontal ones. That's a total of $n(n+1) + 2n = n(n+3).$ Check: $n = 3$ gives $18,$ as it should. Now solve $n(n+3) = 180.$ This factors as $(n-12)(n+15) = 0,$ so $n = 12.$ Therefore, the answer is C.

2018 AMC 10B 第 8 题

答案：C

难度评级：1200

解答：

其他年份的第 8 题