Inductive & Deductive Reasoning
Inductive Reasoning: reasoning from patterns based on the analysis of specific cases. You can
NOT prove a conjecture using algebraic reasoning. Often times you will use inductive reasoning to
come up with a conjecture.
Conjecture: In mathematics, a conjecture is a mathematical statement, which appears likely
to be true, but has not been formally proven to be true under the rules of mathematical
logic.
Ex: What is true about the sum of two odd integers?
If you were using inductive reasoning you would add some pairs of odd integers and look for a
pattern.
1+3=4
3+5=8
5 + 7 = 12
7 + 9 = 16
9 + 11 = 20
11 + 13 = 24
All of the sums appear to be even.
So a possible conjecture would be: If you add two odd integers, then the sum will be an even
integer.
 Remember this is only a conjecture, by trying six cases or even trying a thousand cases does
not prove that this will be true for every two odd integers.
Deductive Reasoning: reasoning from facts, definitions, and accepted properties to new
statements using principles of logic. By using correct deductive reasoning the conclusions you
reach are certain. You can prove a conjectures using deductive reasoning.
Note: You should know the following relationships:
Even Integers can be written in the form: 2m, where m is an integer. (Even numbers have a
factor of two.)
Odd Integers can be written in the form: 2n + 1, where n is an integer. (One more than an even
number will always be an odd.)
Consecutive Integers can be written in the form: n, n + 1, n + 2, etc, where n is an integer. (By
adding one more to the previous number you will get the next consecutive integer.)
Consecutive Odd Integers can be written in the form: 2n + 1, 2n + 3, 2n + 5, etc, where n is an
integer. (By adding two more to the previous number you will get the next consecutive odd
integer.)
Consecutive Even Integers can be written in the form: 2n, 2n + 2, 2n + 4, etc, where n is an
integer. (By adding two more to the previous number you will get the next consecutive even
integer.)
Write a deductive proof that proves that the sum of two odd integers is even.
Let a and b be two odd integers such that a = 2n + 1 and b = 2m + 1, where m and n are integers.
(You need to pick different variables when defining a and b, otherwise they would represent the
same odd integer and we want to show that this relationship is true for any two odd integers.) If
you add a and b then you will get a  b  2n  1  2m  1  2n  2m  2  2n  m  1, which is an
even integer. Therefore, if you add two odd integers then the sum will be even.

Ex: What happens when you multiply two even integers?
Use inductive reasoning to form a conjecture (Try some cases):
Conjecture: _____________________________________________________________________________________________________
__________________________________________________________________________________________________________________
Now use deductive reasoning to prove your conjecture:
Ex: What happens when you multiply two odd integers?
Use inductive reasoning to form a conjecture (Try some cases):
Conjecture: _____________________________________________________________________________________________________
__________________________________________________________________________________________________________________
Now use deductive reasoning to prove your conjecture: