EDGE Math 8

Patterns and Inductive Reasoning (1)

You can find the next term by looking for a pattern and then using that pattern to form a conjecture.

What conjecture did you form in the previous paper-folding exercise? If you arrived at the conclusion that the number of nonoverlapping triangles formed is $2^{n},$ where n is the number of folds, then you made a correct conjecture.

Example

Use inductive reasoning to find the next figure in each set.

1.
2.
3.

Solution

The numbers of circles and squares follow the pattern

1, 2^{2}, 3^{2}, 4^{2}

(or 1, 4, 9, 16).

The number of sides of the polygon increases by 1. The same is true for the number of circles inside the polygon.

Two lines intersect in at most one point. Three lines intersect in at most three points. Four lines intersect in at most six points.