FP 1 Chapter 6 Proof By Induction Dr

FP 1 Chapter 6 – Proof By Induction Dr J Frost (jfrost@tiffin. kingston. sch. uk) Last modified: 31 st March 2015

What is Proof by Induction? We tend to use proof by induction whenever we want to show some property holds for all integers (usually positive). Example …

Chapter Overview We will use Proof of Induction for 4 different types of proof: 1 Summation Proofs 2 Divisibility Proofs 3 Recurrence Relation Proofs 4 Matrix Proofs

The Four Steps of Induction For all these types the process of proof is the same: (Step 1 is commonly known as the ‘base case’ and Step 3 as the ‘inductive case’)

Type 1: Summation Proofs 1 Basis Step ? 2 Assumption ? 3 Inductive ? 4 Conclusion ?

More on the ‘Conclusion Step’

Test Your Understanding Edexcel FP 1 Jan 2010 ?

Exercise 6 A All questions except Q 2.

Type 2: Divisibility Proofs 1 Basis Step 2 Assumption ? ? …

Type 2: Divisibility Proofs Method 1 (the textbook’s method, which I don’t like) Find the difference between successive terms and show this is divisible by the number of interest. 3 Inductive ? Method: You need to explicitly factor out 4 to show divisibility by 4. 4 Conclusion ?

Type 2: Divisibility Proofs 3 Inductive ?

Type 2: Divisibility Proofs 2 Assumption 3 Inductive ? ? Using Method 2

Test Your Understanding 1 Basis Step 2 Assumption 3 Inductive ? ? ? 4 Conclusion ?

Exercise 6 B Page 130. All questions.

Type 3: Recurrence Relation Proofs Notice that we’re trying to prove a position-toterm formula from a term-to-term one. 1 Basis Step ? 2 Assumption 3 Inductive ? ? 4 Conclusion ? Method: Sub in your assumption expression into the recurrence.

Test Your Understanding Edexcel FP 1 Jan 2011 ?

Recurrence Relations based on two previous terms 1 Basis Step 2 Assumption 3 Inductive ? ? ? 4 Conclusion ? (Warning: slightly different to before)

Exercise 6 C Page 132. Q 1, 3, 5, 7

Type 4: Matrix Proofs 1 Basis Step ? 2 Assumption 3 Inductive ? ? 4 Conclusion ? Method: As with summation and divisibility proofs, it’s just a case of ‘separating off’ the assumption expression.

Test Your Understanding 1 Basis Step ? 2 Assumption 3 Inductive ? ? 4 Conclusion ?

Exercise 6 D Page 134. All questions.