Difference between revisions of "2003 AMC 10A Problems/Problem 23"
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== Problem == | == Problem == | ||
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have <math>3</math> rows of small congruent equilateral triangles, with <math>5</math> 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 <math>2003</math> small equilateral triangles? | A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have <math>3</math> rows of small congruent equilateral triangles, with <math>5</math> 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 <math>2003</math> small equilateral triangles? | ||
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<math> \mathrm{(A) \ } 1,004,004 \qquad \mathrm{(B) \ } 1,005,006 \qquad \mathrm{(C) \ } 1,507,509 \qquad \mathrm{(D) \ } 3,015,018 \qquad \mathrm{(E) \ } 6,021,018 </math> | <math> \mathrm{(A) \ } 1,004,004 \qquad \mathrm{(B) \ } 1,005,006 \qquad \mathrm{(C) \ } 1,507,509 \qquad \mathrm{(D) \ } 3,015,018 \qquad \mathrm{(E) \ } 6,021,018 </math> | ||
Revision as of 15:17, 5 November 2006
Problem
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have 
rows of small congruent equilateral triangles, with 
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 
small equilateral triangles?
Solution
There are 
small equilateral triangles.
Each small equilateral triangle needs toothpicks to make it.
But, each toothpick that isn't one of the 
toothpicks on the outside of the large equilateral triangle is a side for 
small equilateral triangles.
So, the number of toothpicks on the inside of the large equilateral triangle is 
Therefore the total number of toothpicks is 
