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FP1 Chapter 6 – Proof By Induction. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: 25 th February 2013. 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.

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fp1 chapter 6 proof by induction

FP1 Chapter 6 – Proof By Induction

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 25thFebruary 2013

slide2

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

Show that for all .

Showing it’s true for a few values of is not a proof. We need to show it’s true for ALL values of .

If it’s true for , it must be true for .

If it’s true for , it must be true for .

If it’s true for , it must be true for .

Suppose we could prove the above is true for the first value of , i.e. .

And so on. Hence the statement must be true for all .

Suppose that if we assumed it’s true for , then we could prove it’s true for . Then:

slide3

Chapter Overview

We will use Proof of Induction for 4 different types of proof:

Show that for all .

1

Summation Proofs

Prove that is divisible by 3 for all .

2

Divisibility Proofs

3

Recurrence Relation Proofs

Given that , prove that

Prove that for all .

4

Matrix Proofs

Bro Tip: Recall that is the set of all integers, and is the set of all positive integers. Thus .

slide4

The Four Steps of Induction

For all these types the process of proof is the same:

! Proof by induction:

Step 1: Basis: Prove the general statement is true for .

Step 2: Assumption:Assume the general statement is true for .

Step 3: Inductive: Show that the general statement is then true for .

Step 4: Conclusion: The general statement is then true for all positive integers .

(Step 1 is commonly known as the ‘base case’ and Step 3 as the ‘inductive case’)

slide5

Type 1: Summation Proofs

Step 1: Basis: Prove the general statement is true for .

Step 2: Assumption: Assume the general statement is true for .

Step 3: Inductive: Show that the general statement is then true for .

Step 4: Conclusion: The general statement is then true for all positive integers .

Show that for all .

When ,

. so summation true for .

1 Basis Step

?

2 Assumption

Assume summation true for .

So

?

When :

Hence true when .

?

3 Inductive

As result is true for , this implies true for all positive integers and hence true by induction.

4 Conclusion

?

slide6

More on the ‘Conclusion Step’

“As result is true for , this implies true for all positive integers and hence true by induction.”

  • I lifted this straight from a mark scheme, hence use this exact wording!
  • The mark scheme specifically says:
  • (For method mark)
  • Any 3 of these seen anywhere in the proof:
  • “true for ”
  • “assume true for ”
  • “true for ”
  • “true for all /positive integers”
slide7

Test Your Understanding

Edexcel FP1 Jan 2010

?

slide8

Exercise 6A

All questions except Q2.

slide9

Type 2: Divisibility Proofs

Step 1: Basis: Prove the general statement is true for .

Step 2: Assumption: Assume the general statement is true for .

Step 3: Inductive: Show that the general statement is then true for .

Step 4: Conclusion: The general statement is then true for all positive integers .

Prove by induction that is divisible by 4 for all positive integers .

Let , where .

which is divisible by 4.

?

1 Basis Step

?

2 Assumption

Assume that for , is divisible by 4 for .

Therefore is divisible by 4 when .

?

3 Inductive

Bro Tip: Putting in terms of allows us to better compare with the assumption case since the power is now the same.

Bro Tip: For divisibility proofs, find the difference, and show that’s divisible by what we’re interested in. This is similar to how for summation proofs we isolated the difference in the sum relative to the previous summation.

4 Conclusion

?

Since true for and if true for , true for for all , therefore true for all integers.

slide10

Test Your Understanding

Step 1: Basis: Prove the general statement is true for .

Step 2: Assumption: Assume the general statement is true for .

Step 3: Inductive: Show that the general statement is then true for .

Step 4: Conclusion: The general statement is then true for all positive integers .

Prove by induction that is divisible by 3 for all positive integers .

Let , where .

which is divisible by 3.

1 Basis Step

?

2 Assumption

Assume that for , is divisible by 3 for .

?

Therefore is divisible by 3 when .

3 Inductive

?

4 Conclusion

?

Since true for and if true for , true for for all , therefore true for all integers by induction.

slide11

Exercise 6B

Page 130.

All questions.

slide12

Type 3: Recurrence Relation Proofs

Step 1: Basis: Prove the general statement is true for .

Step 2: Assumption: Assume the general statement is true for .

Step 3: Inductive: Show that the general statement is then true for .

Step 4: Conclusion: The general statement is then true for all positive integers .

Notice that we’re trying to prove a position-to-term formula from a term-to-term one.

Given that and , prove by induction that

When , as given.

IMPORTANT NOTE: The textbook gets this wrong (bottom of Page 131), saying that is required. I checked a mark scheme – it isn’t.

?

1 Basis Step

2 Assumption

Assume that for , is true for .

?

Then

3 Inductive

?

?

4 Conclusion

Since true for and if true for , true for for all , therefore true for all integers.

This type of proof is slightly different to a summation proof, because we’re using some given statement (the recurrence) to prove another, whereas before we only had one statement.

slide13

Test Your Understanding

Edexcel FP1 Jan 2011

?

slide14

Recurrence Relations based on two previous terms

Step 1: Basis: Prove the general statement is true for .

Step 2: Assumption: Assume the general statement is true for .

Step 3: Inductive: Show that the general statement is then true for .

Step 4: Conclusion: The general statement is then true for all positive integers .

Given that and , , prove by induction that

When , as given.

When , as given.

We have two base cases so had to check 2 terms!

1 Basis Step

?

2 Assumption

Assume that for and , and is true for .

?

We need two previous instances of in our assumption case now.

3 Inductive

Then

?

?

4 Conclusion

Since true for and and if true for and , true for for all , therefore true for all integers.

slide15

Exercise 6C

Page 132.

Q1, 3, 5, 7

slide16

Type 4: Matrix Proofs

Step 1: Basis: Prove the general statement is true for .

Step 2: Assumption: Assume the general statement is true for .

Step 3: Inductive: Show that the general statement is then true for .

Step 4: Conclusion: The general statement is then true for all positive integers .

Prove by induction that for all .

When ,

. As , true for .

1 Basis Step

?

Assume true , i.e.

2 Assumption

?

When ,

Therefore true when .

3 Inductive

?

?

4 Conclusion

Since true for and if true for , true for for all , therefore true for all integers by induction.

slide17

Test Your Understanding

Step 1: Basis: Prove the general statement is true for .

Step 2: Assumption: Assume the general statement is true for .

Step 3: Inductive: Show that the general statement is then true for .

Step 4: Conclusion: The general statement is then true for all positive integers .

Prove by induction that for all .

?

When ,

. As , true for .

1 Basis Step

Assume true , i.e.

?

2 Assumption

When ,

Therefore true when .

?

3 Inductive

?

4 Conclusion

Since true for and if true for , true for for all , therefore true for all integers by induction.

slide18

Exercise 6D

Page 134.

All questions.