AMC10 2018 B

AMC10 2018 B · Q8

AMC10 2018 B · Q8. It mainly tests Quadratic equations, Sequences & recursion (algebra).

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?

Sara用牙签搭建了一个楼梯，如图所示：[这是一个3级楼梯，使用了18根牙签。] 使用180根牙签的楼梯有多少级？

(A) 10 10

(B) 11 11

(D) 24 24

(E) 30 30

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): In the staircase with $n$ steps, the number of vertical toothpicks is $1+2+3+\cdots+n+n=\dfrac{n(n+1)}{2}+n.$ There are an equal number of horizontal toothpicks, for a total of $n(n+1)+2n$ toothpicks. Solving $n(n+1)+2n=180$ with $n>0$ yields $n=12$.

答案（C）：在有 $n$ 级台阶的楼梯中，竖直牙签的数量为 $1+2+3+\cdots+n+n=\dfrac{n(n+1)}{2}+n.$ 水平牙签的数量与竖直牙签相同，因此牙签总数为 $n(n+1)+2n$。在 $n>0$ 的条件下解方程 $n(n+1)+2n=180$，得到 $n=12$。

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