Further Pure 1

1
Further Pure 1

• Lesson 11
• Proof by induction

2
Mathematical Statements

• In this lesson we are going to look at proving
  mathematical statements.
• For example 3n-1 is always an even number
• The problem is though, how can we prove this to
  be true as there are infinitely many values of n
  to try?
• We can see that if n 1, then 31-1 2 Even
• and if n 2, then 32-1 8 Even
• and if n 3, then 33-1 26 Even
• But it is impossible to test every single value
  of n to see if the statement is true.
• We are going to now look at the idea of proof by
  induction.

3
Proof by induction

• Here is an example to think about.
• An old woman is asked how old she is, and replies
  I am 100 years old. No record has been kept of
  her birthday so how do we now she is 100?
• Answer Because she was 99 last year.
• With proof by induction we assume that a
  statement is true and then try to prove that the
  next result will also be true.
• If this principle holds then as long as the first
  value works then automatically so must the
  second, then the third and so on and so on.
• Its a bit like pushing over a domino line.

4
Proof by induction

• Now we can go back to our original problem from
  the first slide.
• Prove that 3n-1 is always an even number
• Lets assume that 3k-1 is always even, for n k.
• Now 3k1-1 33k-1
• 33k-3 2
• 3(3k-1) 2
• 3(Even) Even
• Even
• So we have shown that if 3n-1 is even for n k
  then 3n-1 will be even for n k1.
• Now if k 1, 3k-1 Even
• The domino effect kicks in and it must be true
  for n 2,3,8

5
Questions

• 1) Prove that, for all positive integers n,
• 1 2 3 n (n/2)(n1)
• 2) Prove that, for all positive integers n,
• 12 22 32 n2 (n/6)(n1)(2n1)
• 3) Prove that, for all positive integers n,
• 13 23 33 n3 (n2/4) (n1)2
• These are important results and we shall learn
  some more about them in lesson 12

6
Solution 1

• Prove that, for all positive integers n,
• 1 2 3 n (n/2)(n1)
• First we start by assuming that it is true for n
  k and try to show that it will be true for n
  k1
• This shows that if n k works then n k1 will
  also work.
• So if n 1, then the sum of 1 is 1 0.512 1
  which is the sum of 1.
• Therefore by induction the original statement is
  true for all n.

7
Solution 2

• Prove that, for all positive integers n,
• 12 22 32 n2 (n/6)(n1)(2n1)
• First we start by assuming that it is true for n
  k and try to show that it will be true for n
  k1
• This shows that if n k works then n k1 will
  also work.
• So if n 1, then the sum of 1 is 1
  (1/6)123 1 which is the sum of 1.
• Therefore by induction the original statement is
  true for all n

8
Solution 3

• Prove that, for all positive integers n,
  13 23 33 n3 (n/4)2(n1)2
• First we start by assuming that it is true for n
  k and try to show that it will be true for n
  k1
• This shows that if n k works then n k1 will
  also work.
• So if n 1, then the sum of 1 is 1 0.254 1
  which is the sum of 1.
• Therefore by induction the original statement is
  true for all n

9
More Proofs by induction 1

• A sequence is defined by un1 4un 3, u1 2
• Prove that un 4n-1 1
• We start by assuming that n k will work and
  investigate what happens with n k1
• uk1 4uk 3
• 4(4k-1 1) 3
• 44k-1 4 3
• 4k 4 3
• 4k 1

10
More Proofs by induction 2

• Prove that un 4n 6n 1 is divisible by 9 for
  all n gt 0.
• uk1 4k1 6(k1) 1
• 4k1 6k 6 1
• 4k1 6k 5
• 44k 24k 18k 4 4 5
• With this line I have just added in values so
  that I can factorise to make un
• (44k 24k 4) 18k 4 5
• 4(4k 6k -1) 18k 9
• 4(uk) 9(2k 1)
• This proves that uk1 is divisible by 9 because
  uk is divisible by 9 and the second part 9(2k
  1) must also be divisible by 9.
• Now if n 1, then u1 41 61 1 9 which is
  divisible by 9.
• This proves that un must always be divisible by 9
  for all n gt 0.

11
Enrichment problem

• Let P(n) be the statement that
• is divisible by 44.
• I shall show this is true if n is any positive
  integer!
• If n1 then is equal
  to 44
• so is divisible by 44!
• So the statement is clearly true when n1.

12
Assume statement is true for nk

• We suppose that the statement is true for nk
  where k is some positive integer.
• So we assume that
  is divisible by 44.
• We want to then show that the statement must also
  be true for nk1.

13
Considering nk1

• We therefore want to consider
• (this is with nk1).
• We want to show this is divisible by 44.
• All we are allowed to assume is that
• Is divisible by 44 (the case nk).
• So lets see if we can express
  in terms of .

14
The trick!

• So if is divisible by 44
  with nk then it is also divisible by 44 with
  nk1.

15
The conclusion

• So we have shown that is
  divisible by 44 with n1.
• If it is divisible by 44 with nk, for any
  positive integer k, then it follows that it is
  divisible by 44 with nk1.
• Therefore it follows that
  must be divisible by 44 for all positive
  integers n!