1. Mathematical Induction
By: Emily Elrod and Lora Morris

2. History 1623 – 1662 2nd to use “induction” More modern proof
Francesco Maurolico
1494 – 1575
First to use “induction”
Sketchy Proof
Published Works
Blaise Pascal
1623 – 1662
2nd to use “induction”
More modern proof

3. Maurolico’s Proof1
By Proposition 13 [this proposition asserts that n2 + (2n+1) = (n + 1)2] the first square number 1 added to the following odd number 3 makes the following square number 4; and this second square number 4 added to the third odd number 5 makes the third square number 9; and likewise the third square number 9 added to the fourth odd number 7 makes the fourth square number 16; and so successively to infinity the result is demonstrated by repeated application of Proposition 13.
1. Burton, 464

4. Pascal’s Proof2 In modern notation, Pascal proved
Although this proposition has an infinite number of cases, I will give a rather short demonstration by supposing two lemmas.
The first one, which is self-evident, is that this proposition occurs in the second base; because it is apparent that  is to  as 1 is to 1 [that is, for n=1,
The second one is that if this proposition is true for an arbitrary base, it will necessarily be true in the next base. From which it is clear that it will necessarily be true in all bases, because it is true in the second base by the first lemma; hence by means of the second lemma, it is true in the third base hence in the fourth base, and so on to infinity.
It is therefore necessary only to prove the second lemma.
2. Burton,

5. Was it always called ‘mathematical induction’?
John Wallis
1616 – 1703
placed a name on this type of logical thinking: “reasoning by recurrence”
Augustus De Morgan
1806 – 1871
First to use the terminology ‘mathematical induction’ (1838)

6. Flaw For every Proof: Induction. Assume for some k that Then
Hence the formula holds for k + 1.
Conclusion. Therefore, by the principle of mathematical induction, the formula holds for all

7. What Does an Induction Proof have to have?
1. Proposition
2. Basis for the proof, where we determine if P(1) is true.
3. Induction, where we suppose the formula holds for k and consider k+1.
4. Conclusion

8. The type of proofs we encountered
Summation
Ex. Find and prove the formula for
Inequality
Prove the inequality for appropriate choice of n.
Divisibility
Prove that , for all n ≥ 1, is divisible by 6.

9. Exercises for the class.
Work together to prove the following by Induction. Write what you have on the board for the last two minutes. You have a total of 10 minutes to work.
#1 FRONT ROW Prove the inequality for appropriate choice of n.
# 2 BACK ROW Prove that, for all n ≥ 1, is divisible by 6.

10. Exercise #1

11. Exercise #2

12. Challenging Exercise
Prove that, for all , is divisible by 5.

13. Challenging Exercise

14. Any Questions?

15. References
Burton, D.M. (2007). The history of mathematics: an introduction. New York, NY: McGraw Hill.