2003 AMC 10A Problem 23

Problem 23 of 25HarderAlgebra

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?

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Solution

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.

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