Honors Geometry Section 1.0

Patterns and Inductive Reasoning

Geometry, like much of mathematics and science,

developed when people began recognizing and describing

patterns. Much of the reasoning in geometry consists of three

steps.

1.

2.

3.

Recognize a pattern.

Make a conjecture about the pattern.

A conjecture is an educated guess based on past observations.

Prove the conjecture.

Example 1: Give the next two terms in each sequence of numbers and describe the pattern in words.

2, 6, 18, 54… 1 1 1

1, , , ,...2 4 8

486 ,162

3by Multiply 32

1- ,161

21-by Multiply

Example 1: Give the next two terms in each sequence of numbers and describe the pattern in words.

1, 3, 5, 7, 9…

1, 1.1, 21.1, 21.12, 321.12, …

13 ,11 2 Add

4321.123 ,123.321rightfar then theandleft far

theinteger tolarger next theAdd

A sequence can be specified by an equation or “rule”. For the first

example (2,6,18,54,…), the sequence can be specified by the rule

where n = 1,2,3, etc. corresponding to the 1st term, 2nd term, 3rd term ,

etc.

12 3nna

Example 2: Write a rule for the nth term for the 2nd and 3rd sequences in example 1.

1, 3, 5, 7, 9

1 1 11, , , ,...

2 4 8 1

211

n

na

)1(21 nan

Reasoning based on past observations is called inductive reasoning.

Keep in mind that inductive reasoning does not guarantee a correct conclusion.

Later in the course, we will prove a conjecture is true using deductive reasoning. To prove a conjecture is

false, you need to show a single example where the conjecture is

false. This single example is called acounterexample.

Example 2: Show the conjecture is false.

The product of two positive numbers is always greater than the larger number.

If m is an integer*, then m2 > 0.

771 5.372

1

002

You should know the following sets of numbers:

whole numbers:

integers:

rational numbers:

irrational numbers:

,...3,2,1,0

,...2,1,0,1,2...,

integers are b and a and b

a as written becan

rationalnot is