Discrete Structures Chapter 5: Sequences, Mathematical Induction, and Recursion 5.2 Mathematical Induction I [Mathematical induction is] the standard proof technique in computer science. – Anthony Ralston 5.2 Mathematical Induction I

5.2 Mathematical Induction I Introduction Mathematical Induction is one of the more recently developed techniques of proof in the history of mathematics. It is used to check conjectures about the outcomes of processes that occur repeatedly an according to definite patterns. 5.2 Mathematical Induction I

5.2 Mathematical Induction I Note Please make sure that you read through the proofs and examples in the text book. We will be doing different problems in class so that you will have more examples for reference. The more you practice, the easier induction becomes. 5.2 Mathematical Induction I

Method of Proof by Mathematical Induction Consider a statement of the form For all integers n  a, a property P(n) is true. Step 1 (basic Step): Show that P(a) is true. Step 2 (inductive Step): Assume that P(k) is true for all integers k  a. (inductive hypothesis) Show that P(k+1) is true 5.2 Mathematical Induction I

5.2 Mathematical Induction I Proposition Proposition 5.2.1 For all integers n  8, n¢ can obtained using 3¢ and 5¢ coins. 5.2 Mathematical Induction I

5.2 Mathematical Induction I Theorems Theorem 5.2.2 Sum of the First n Integers For all integers n  1, Theorem 5.2.3 Sum of Geometric Sequence For any real number r except 1, and any integer n  0, 5.2 Mathematical Induction I

5.2 Mathematical Induction I Definition Closed Form If a sum with a variable number of terms is shown to be equal to a formula that does not contain either an ellipsis or a summation symbol, we say that it is written in closed form. Closed Form 5.2 Mathematical Induction I

5.2 Mathematical Induction I Example – pg. 257 #7 Prove each statement using mathematical induction. Do not derive them from Theorems 5.2.2 or 5.2.3. 5.2 Mathematical Induction I

5.2 Mathematical Induction I Examples – pg. 257 Prove each statement by mathematical induction. 5.2 Mathematical Induction I

5.2 Mathematical Induction I Examples – pg. 257 Use the formula for the sum of the first n integers and/or the formula for the sum of a geometric sequence to evaluate the sums or to write them in closed form. 5.2 Mathematical Induction I