You may want to find the tenth or the one-hundreth term in a sequence. In this case, rather than find every previous term, you can look for a pattern and make a conjecture. A conjecture is a conclusion you reach using inductive reasoning.
Problem 2 Using Inductive Reasoning
Look at the circles. What conjecture can you make about the number of regions 20 diameters form?
Plan
Do you need to draw a circle with 20 diameters?
No. Solve a simpler problem by finding the number of regions formed by 1, 2, and 3 diameters. Then look for a pattern.
Each circle has twice as many regions as diameters. Twenty diameters form
✓ Got It?
- What conjecture can you make about the twenty-first term in R, W, B, R, W, B, …?
It is important to gather enough data before you make a conjecture. For example, you do not have enough information about the sequence 1, 3,…to make a reasonable conjecture. The next term could be
Problem 3 Collecting Information to Make a Conjecture
What conjecture can you make about the sum of the first 30 even numbers?
Plan
What's the first step?
Start by gathering data. You can organize your data by making a table.
Find the first few sums and look for a pattern.
| Number of Terms | Sum | |
|---|---|---|
| 1 |
|
|
| 2 |
|
|
| 3 |
|
|
| 4 |
|
You can conclude that the sum of the first 30 even numbers is
✓ Got It?
- What conjecture can you make about the sum of the first 30 odd numbers?
Table of Contents
- 6-1 The Polygon Angle-Sum Theorems
- 6-2 Properties of Parallelograms
- 6-3 Proving That a Quadrilateral Is a Parallelogram
- 6-4 Properties of Rhombuses, Rectangles, and Squares
- 6-5 Conditions for Rhombuses, Rectangles, and Squares
- 6-6 Trapezoids and Kites
- 6-7 Polygons in the Coordinate Plane
- 6-8 and 6-9 Coordinate Geometry and Coordinate Proofs