Question

a. describe the pattern below. b. draw the next 2 figures for this sequence.

Explanation:

Step1: Analyze the number of triangles

The first figure has 1 triangle. The second figure has 4 triangles (a 2 - layer triangular - like structure), and the third figure has 9 triangles (a 3 - layer triangular - like structure). The fourth figure has 16 triangles (a 4 - layer triangular - like structure), and the fifth figure has 25 triangles (a 5 - layer triangular - like structure). The number of triangles in each figure is the square of the figure number \(n\), i.e., the number of triangles \(T(n)=n^{2}\). Also, as noted in the description, each new bottom - row has two more triangles than the previous bottom - row.

Step2: Draw the next figures

For the next figure (\(n = 6\)), it will be a 6 - layer triangular - like structure. The bottom - row will have \(2\times6 - 1=11\) triangles (using the arithmetic - sequence property for the number of triangles in the bottom - row: \(a_n=a_1+(n - 1)d\), where \(a_1 = 1\), \(d = 2\)). The total number of triangles will be \(6^{2}=36\). For the figure after that (\(n = 7\)), it will be a 7 - layer triangular - like structure with \(7^{2}=49\) triangles and the bottom - row having \(2\times7 - 1 = 13\) triangles.

Answer:

The pattern is that the number of triangles in the \(n\)th figure is \(n^{2}\), and each new bottom - row has two more triangles than the previous bottom - row. The next two figures should be drawn as 6 - layer and 7 - layer triangular - like structures with 36 and 49 triangles respectively.