5.2 Mathematical Induction I 1

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 2

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 3

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 4

Method of Proof by Mathematical Induction

• Consider a statement of the formFor 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

Proposition

• Proposition 5.2.1

For all integers n 8, n¢ can obtained using 3¢ and 5¢ coins.

5.2 Mathematical Induction I 6

Theorems

• Theorem 5.2.2 Sum of the First n Integers

For all integers n 1,

• Theorem 5.2.3 Sum of Geometric SequenceFor any real number r except 1, and any integer n 0,

11 2 3 ...

2

n nn

1

0

1

1

nni

i

rr

r

5.2 Mathematical Induction I 7

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.

11 2 3 ...

2

n nn

Closed Form

5.2 Mathematical Induction I 8

Example – pg. 257 #7

• Prove each statement using mathematical induction. Do not derive them from Theorems 5.2.2 or 5.2.3.

For all integers 1,

5 3 1 6 11 16 ... 5 4 .

2

n

n nn

5.2 Mathematical Induction I 9

Examples – pg. 257

• Prove each statement by mathematical induction.

2

3 3 3

12

1

111. 1 2 +... = , for all integers 1.

2

14. 2 2 2, for all integers 0.n

i n

i

n nn n

i n n

5.2 Mathematical Induction I 10

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.

2 3

21. 5 10 15 20 ... 300

26. 3 3 +3 ... 3 , where is an integer with 1.n n n