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 a n with recursive description a n = a n-1 + (2n – 1) for n ≥ 2 and a 1 = 1. a 1 = ____ a 2 =____ + ____ = ____ a 3 =____ + ____ = ____ a 4 =____ + ____ = ____ a 5 =____ + ____ = ____ :: : :

5. A different setting If you are told that a 23 = 529, can you find a 24 using only the recursive pattern? Does this agree with the proposed closed formula?

6. The recursive sequence We wish to prove that the sequence a n with recursive description a n = a n-1 + (2n – 1) for n ≥ 2 and a 1 = 1 has the closed formula a n = n 2 for all n ≥ 1.

7. Picturing mathematical induction We are given the sequence a n with recursive description a n = a n-1 + (2n – 1) for n ≥ 2 and a 1 = 1, and we will verify the following sequence of statements in order:  a 1 = 1 2  a 2 = 2 2  a 3 = 3 2  a 4 = 4 2  a 5 = 5 2  … etc …

8. Picturing mathematical induction The following table form is handy for keeping the correct mental image: na n (rec. def.)n2n2 Equal? 1a 1 = 2a 2 = 3a 3 = 4a 4 = :::: m-1a m-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 n=1. Let’s 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 a n with recursive description a n = a n-1 + (2n – 1) for n ≥ 2 and a 1 = 1 has the closed formula a n = n 2 for all n ≥ 1. Proof by induction. The statement “a 1 = 1 2 ” is true because a 1 = 1 by its definition. Now suppose that statements “a 1 = 1 2 ”, “a 2 = 2 2 ”, “a 3 = 3 2 ”, …, “a m-1 = (m-1) 2 ” have all been checked to be true for some integer m ≥ 2, and let’s consider the next statement: a m = a m-1 + (2m-1) by definition = (m-1) 2 + (2m-1) by previously checked = (m 2 - 2m + 1) + (2m-1) by algebra = m 2 by more algebra This establishes that the next statement, “a m = m 2,” 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: nSum n terms2 n+1 – 2Equal? 1 2121 2 2 – 2Yes 2 2 1 + 2 2 2 3 – 2Yes 3 2 1 + 2 2 + 2 3 2 4 – 2Yes 4 2 1 + 2 2 + 2 3 + 2 4 2 5 – 2Yes : ::: m-1 2 1 + 2 2 + 2 3 +…+ 2 m-1 2 m – 2Yes 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.