Question

用牙签排成若干行小等边三角形，从而构成一个大等边三角形。例如图中有 $3$ 行全等小等边三角形，底行有 $5$ 个小三角形。如果大等边三角形的底行由 $2003$ 个小等边三角形组成，需要多少根牙签？

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles?

$1{,}004{,}004$

$1{,}005{,}006$

$1{,}507{,}509$

$3{,}015{,}018$

$6{,}021{,}018$

答案：C

知识点：求和找规律

难度评级：1730

解答：

有 $n$ 行时，底行小三角形个数为 $2n - 1$ ，所以 $2n - 1 = 2003$ ，得 $n = 1002$ 。

第 $k$ 行需要 $3k$ 根牙签，所以总数为 $3(1 + 2 + \cdots + 1002)$ 。

这等于 $3 \cdot \dfrac{1002 \cdot 1003}{2} = 1{,}507{,}509$ 。

所以正确答案是 C。

A triangle with $n$ rows has $2n - 1$ small triangles in its base row, so $2n - 1 = 2003$ gives $n = 1002.$

Each row $k$ requires $3k$ toothpicks, so the total is $3(1 + 2 + \cdots + 1002).$

This equals $3 \cdot \dfrac{1002 \cdot 1003}{2} = 1{,}507{,}509.$

Thus, the correct answer is C.

2003 AMC 10A 第 23 题

答案：C

难度评级：1730

解答：

其他年份的第 23 题