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) 6 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) . 6