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
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