Mathematical Induction and Proofs: Chapter 2b, Study notes of Mathematics

Download Mathematical Induction and Proofs: Chapter 2b and more Study notes Mathematics in PDF only on Docsity! MthT 430 Notes Chapter 2b Induction etc. Proofs by Mathematical Induction PMI is also stated as Suppose P (n) is a statement for each natural number n. If { P (1) is true, Whenever P (k) is true, P (k + 1) is true. then P (n) is true for all n ∈ N. A proof using PMI requires: • Carefully stating the proposition or statement P (k), which may be a sentence (paragraph) or equation, inequality, . . ., which depends on k. • Proving that P (1) is true – usually a simple verification. • Assume P (k)1 – write down the statement or formula P (k): • Assume . . . (statement or formula which depends on k). • Apply valid operations (theorems, add to both sides of an equation, etc.) . . . Remember that P (1) is valid too. Try to arrive at P (k + 1) as a statement or equation. Examples 1. Prove the formula 12 + 22 + . . . + n2 = n(n + 1)(2n + 1) 6 . • P (n) is the sentence 12 + 22 + . . . + n2 = n(n + 1)(2n + 1) 6 . What is the verb of the sentence? 1 As a variant, assume that P (1), . . . , P (k), are true and use PCI. chap2b.pdf page 1/4 • P (1) is true means to verify the statement 12 = 1(1 + 1)(2 · 1 + 1) 6 . • P (n) is true implies P (n + 1) is true means to assume 12 + 22 + . . . + n2 = n(n + 1)(2n + 1) 6 , and to show that 12 + 22 + . . . + n2 + (n + 1)2 = (n + 1) ((n + 1) + 1)(2 (n + 1) + 1) 6 The valid operations are adding (n + 1)2 to both sides of the equation P (n) and various algebraic manipulations. 2. Tower of Hanoi • P (n) is the statement: • n rings can be moved from one spindle to another with 2n − 1 moves, and no fewer. Other manageable forms of P (n) are • The minimal number of moves to move n rings from one spindle to another is 2n − 1. • It takes 2n − 1 moves to move n rings from one spindle to another. • P (1) is easy. • P (n) is true implies P (n + 1) is true means to assume n rings can be moved from one spindle to another with 2n − 1 moves, and no fewer , and to show that n + 1 rings can be moved from one spindle to another with 2n+1 − 1 moves, and no fewer . In this case, the valid operations are a precise sequence of moves to move the n + 1 rings – the top n rings are moved using P (n), the bottom ring is moved and then the top rings are moved a second time using P (n). chap2b.pdf page 2/4