Mathematical Induction

1
Section 2.3

• Mathematical Induction

2
First Example

• Investigate the sum of the first n positive odd
  integers.
• 1 ____
• 1 3 ____
• 1 3 5 ____
• 1 3 5 7 ____
• 1 3 5 7 9 ____

3
First Example

• Suppose that some diligent person has carefully
  determined that
• 1 3 5 7 29 31 256
• What is the value of 1 3 5 7 31 33?
• What is the value of 1 3 5 7 33 35?

4
A different setting

• Compare this to the sequence an with recursive
  description an an-1 (2n 1) for n 2 and a1
  1.
• a1 ____
• a2 ____ ____ ____
• a3 ____ ____ ____
• a4 ____ ____ ____
• a5 ____ ____ ____

5
A different setting

• If you are told that a23 529, can you find a24
  using only the recursive pattern?
• Does this agree with the proposed closed formula?

6
The recursive sequence

• We wish to prove that the sequence an with
  recursive description an an-1 (2n 1) for n
  2 and a1 1
• has the closed formula an n2 for all n 1.

7
Picturing mathematical induction

• We are given the sequence an with recursive
  description an an-1 (2n 1) for n 2 and
  a1 1,
• and we will verify the following sequence of
  statements in order
• a1 12
• a2 22
• a3 32
• a4 42
• a5 52
• etc

8
Picturing mathematical induction

• The following table form is handy for keeping the
  correct mental image

n an (rec. def.) n2 Equal?
1 a1
2 a2
3 a3
4 a4

m-1 am-1
m
9
Reader vs. Author Proof.

• The Author tries to convince the reader the
  result holds by showing that it holds for
  successive values of n, beginning with n1.
• Lets all agree to call the last row that you
  verified the m-1th row. Therefore the Author
  must convince the Reader that the result holds
  for the next row, which is, the mth row.

10
The formal proof

• Claim. The sequence an with recursive description
  an an-1 (2n 1) for n 2 and a1 1 has the
  closed formula an n2 for all n 1.
• Proof by induction. The statement a1 12 is
  true because a1 1 by its definition.
• Now suppose that statements a1 12, a2
  22, a3 32, , am-1 (m-1)2 have all been
  checked to be true for some integer m 2, and
  lets consider the next statement am
  am-1 (2m-1) by definition
• (m-1)2 (2m-1) by previously checked
• (m2 - 2m 1) (2m-1) by algebra
• m2 by more algebra
• This establishes that the next statement, am
  m2, is true, completing the induction argument.

11
Formulas for sums

• Investigate the sum of the first n powers of 2
• 2 ____
• 2 4 ____
• 2 4 8 ____
• 2 4 8 16 ____
• In general,

12
Closed formula for a sum

• We wish to prove that for all n 1,

13
Picturing mathematical induction

• The following table form is handy for keeping the
  correct mental image

n Sum n terms 2n1 2 Equal?
1 21 22 2 Yes
2 21 22 23 2 Yes
3 21 22 23 24 2 Yes
4 21 22 23 24 25 2 Yes

m-1 21 22 23 2m-1 2m 2 Yes
m
14
The formal proof

• Claim. For all n 1,
• Proof by induction. The statement
  is true.
• Now let m 2 be the first summation not yet
  checked. This means
• that the statement has been checked
• So
• This establishes that the next statement,
  is true.