2003 AMC 10A Problem 23

Question

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

Answer: C

Concepts:summation pattern recognition

Difficulty rating: 1730

Small hint:

A base row of

2003

small triangles means

2n - 1 = 2003,

so there are

n

rows

Big hint:

Row

k

needs

3k

toothpicks, so the total is

3(1 + 2 + \cdots + n)

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