13.4 Mathematical Induction

Mathematical Induction is a common method of proving that each statement of an infinite sequence of mathematical statements is true. The Principle of Math Induction contains BOTH English & mathematical statements in three steps. *You will lose points if you don’t include the English statements for each step! Follow the pattern! Let P n be a statement involving natural numbers, then I. P 1 Prove that P 1 is true for n = 1 (Show mathematically it works for n = 1) II. P k Assume P k is true for n = k. (There is no proof here, truth of one statement implies truth of the next statement) III. P k+1 Prove that P k+1 is true for n = k + 1 (This statement builds from step II) Conclusion: “By mathematical induction, the above statement is true for all natural numbers.”

Ex 1) Use mathematical induction to prove that the sum of the first n odd positive integers is n … + (2n – 1) = n 2 I. Prove that P 1 is true for n = 1 Does 1 = (1) 2 ? 1 = 1 · 1 1 = 1 II. Assume P k is true for n = k … + (2k – 1) = k 2 III. Prove that P k+1 is true for n = k … + (2k – 1) = k 2 (from step II) … + (2k – 1) + (2(k + 1) – 1) = k 2 + (2(k + 1) – 1) = k 2 + (2k + 2 – 1) = k 2 + 2k + 1 = (k + 1) 2 By mathematical induction, the above statement is true for all natural numbers. Q.E.D. add next term Latin for quod erat demonstrandum meaning “which had to be demonstrated” We want RHS = (k + 1) 2

Homework #1304 Pg 702 #5–9 odd, 12–16 all, 21, 23 *Homework Hints* #5,7: You are just finding P k+1 – plug in (k + 1) for each n on the right hand side and simplify! #9: Interpret sentence into math symbols / equations – that’s it! #12–16, 21, 23: These are full-on mathematical induction problems! Remember Laws of Exponents: 3 2 ·3 4 = So…3·3 k = 3 k+1 3·3 k+1 = 3 k+2 For extra practice, let’s do a homework question together.

HW #12) Use mathematical induction to prove for all positive integers n. I. Prove that P 1 is true for n = 1 II. Assume P k is true for n = k.

III. Prove that P k+1 is true for n = k + 1 By mathematical induction, the above statement is true for all natural numbers. Q.E.D. We want RHS to (Factor out k+1 first)