Chapter 1, Problem 26CT

Solution

To describe the tenth figure in the pattern.

Given:

The patterns:

  

Calculation:

Note that the first figure has 1 triangle. The second figure has 4 small triangles in it, so it can be expressed as the square of 2, that is, $2^2=4$ .

The third figure has 9 small triangles in it, so the number of small triangles in the third figure can be expressed as the square of 3, that is, $3^2=9$ .

And the fourth figure has 16 small triangles in it, so the number of small triangles in the third figure can be expressed as the square of 4, that is, $4^2=16$ .

Thus, by following the same procedure, the tenth figure must have ${10}^2=100$ small triangles in it.