Count the Triangles I

by
Angie, Beth, and Teisha

Count the triangles. Extend the sequence.

The sequence here is , where n equals the number of sections into which the two lateral sides of the triangle may be divided.

In the triangle shown above, only one triangle is constructed. .

In the second triangle, 8 smaller triangles may be developed. Each lateral side is divided into two sections.

In the third triangle, 27 smaller triangles may be constructed. Each lateral side is divided into three sections.

In the fourth triangle, 64 smaller triangles may be constructed. Each lateral side is divided into four sections.

The sequence continues in such a manner. From this sequence, one might hypothesize that the following triangle:

which has lateral sides divided into 5 sections, can be divided into sections.
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