Avon High School
Section: 11.4
ACE COLLEGE ALGEBRA II - NOTES
Mathematical Induction
Mr. Record: Room ALC-129
Semester 2 - Days 41 & 42
Eureka! He found it! After ten years of work, Princeton
University’s Andrew Wiles proved Fermat’s Last Theorem.
Pierre de Fermat (1601-1665) was a lawyer who enjoyed studying
mathematics. In a margin of one of his books, he claimed that no
two positive integers satisfy the equation
x n  y n  z n for n greater than or equal to 3.
Fermat claimed to have a proof for this conjecture, but added,
“The margin of my book is too narrow to write it down.”
In 1994, Wiles used a technique called mathematical induction to prove this theorem.
The Principle of Mathematical Induction
How do we prove statements using mathematical induction?
The Principle of Mathematical Induction
Let Sn be a statement involving the positive integer n. If
1. S1 is true, and
2. the truth of the statement Sk implies the truth of the statement Sk+1, for every positive
integer k,
then the statement Sn is true for all positive integers n.
The Steps in a Proof by Mathematical Induction
Let Sn be a statement involving the positive integer n. To prove that Sn is true for all positive
integers n requires two steps.
Step 1: Show that S1 is true.
Step 2: Show that if Sk is assumed to be true, then Sk+1 is also true, for every positive integer k.
Example 1
Writing S1, Sk, and Sk
+1
For the given statement Sn, write the three statements S1, Sk, and Sk+1.
a. Sn: 1  2  3 
n
n(n  1)
2
b. Sn: 12  22  32 
 n2 
n(n  1)(2n  1)
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Proving Statements about Positive Integers Using
Mathematical Induction
Example 2
Proving a Formula by Mathematical Induction
Use mathematical induction to prove 1  2  3 
Example 3
n
n(n  1)
.
2
Proving a Formula by Mathematical Induction
Use mathematical induction to prove 12  22  32 
Example 4
 n2 
Using the Principle of Mathematical Induction
Prove that 2 is a factor of n 2  5n for all positive integers n.
n(n  1)(2n  1)
.
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