1. 11.7 – Proof by Mathematical Induction

2. Mathematical induction is a method of proving statements involving natural numbers.

3. Mathematical induction is a method of proving statements involving natural numbers.
To prove a statement true for all natural numbers, n:

4. Mathematical induction is a method of proving statements involving natural numbers.
To prove a statement true for all natural numbers, n:
1. Show that the statement is true for n = 1

5. Mathematical induction is a method of proving statements involving natural numbers.
To prove a statement true for all natural numbers, n:
1. Show that the statement is true for n = 1
2. Assume that the statement is true for some natural number, k. (induction hypothesis)

6. Mathematical induction is a method of proving statements involving natural numbers.
To prove a statement true for all natural numbers, n:
1. Show that the statement is true for n = 1
2. Assume that the statement is true for some natural number, k. (induction hypothesis)
3. Show that the statement is true for the next natural number, k + 1.

7. Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2 4

8. Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2 4 1
Ex. 1 Prove that … + n3 = n2(n+1) Show that the statement is true for n = 1.

9. Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2 4 1
Ex. 1 Prove that … + n3 = n2(n+1) Show that the statement is true for n = … + n3 = n2(n+1)2

10. Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2 4 1
Ex. 1 Prove that … + n3 = n2(n+1) Show that the statement is true for n = … + n3 = n2(n+1)2 n = 1:

11. Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2 4 1
Ex. 1 Prove that … + n3 = n2(n+1) Show that the statement is true for n = … + n3 = n2(n+1)2 n = 1: 13= 12(1+1)2

12. Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2 4 1
Ex. 1 Prove that … + n3 = n2(n+1) Show that the statement is true for n = … + n3 = n2(n+1)2 n = 1: 13= 12(1+1)2

13. Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2 4 1
Ex. 1 Prove that … + n3 = n2(n+1) Show that the statement is true for n = … + n3 = n2(n+1)2 n = 1: 13= 12(1+1)2 1 = 1(2)2

14. Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2 4 1
Ex. 1 Prove that … + n3 = n2(n+1) Show that the statement is true for n = … + n3 = n2(n+1)2 n = 1: 13= 12(1+1)2 1 = 1(2)2 1 = 1, thus shown to be true.

15. 2. Assume that the statement is true for some natural number, k.

16. 2. Assume that the statement is true for some natural number, k
2. Assume that the statement is true for some natural number, k … + n3 = n2(n+1)2 4

17. 2. Assume that the statement is true for some natural number, k
2. Assume that the statement is true for some natural number, k … + n3 = n2(n+1) … + k3 = k2(k+1)2

18. 3. Show that the statement is true for the next natural number, k + 1.

19. 3. Show that the statement is true for the next natural number, k + 1
3. Show that the statement is true for the next natural number, k … + k3 = k2(k+1)2 4

20. 3. Show that the statement is true for the next natural number, k + 1
3. Show that the statement is true for the next natural number, k … + k3 = k2(k+1) … + k3 + (k+1)3 = k2(k+1)2 + (k+1)3

21. 3. Show that the statement is true for the next natural number, k + 1
3. Show that the statement is true for the next natural number, k … + k3 = k2(k+1) … + k3 + (k+1)3 = k2(k+1)2 + (k+1)3 = k2(k+1)2 + (k+1)3 4 1

22. 3. Show that the statement is true for the next natural number, k + 1
3. Show that the statement is true for the next natural number, k … + k3 = k2(k+1) … + k3 + (k+1)3 = k2(k+1)2 + (k+1)3 = k2(k+1)2 + (k+1)

23. 3. Show that the statement is true for the next natural number, k + 1
3. Show that the statement is true for the next natural number, k … + k3 = k2(k+1) … + k3 + (k+1)3 = k2(k+1)2 + (k+1)3 = k2(k+1)2 + (k+1) = k2(k+1)2 + 4(k+1)3 4 4

24. = k2(k+1)2 + 4(k+1)3 4

25. = k2(k+1)2 + 4(k+1)3 4 = (k+1)2 [k2 + 4(k+1)]

26. = k2(k+1)2 + 4(k+1)3 4 = (k+1)2 [k2 + 4(k+1)] = (k+1)2 [k2 + 4k+4]

27. = k2(k+1)2 + 4(k+1)3 4 = (k+1)2 [k2 + 4(k+1)] = (k+1)2 [k2 + 4k+4] = (k+1)2 (k+2)2 4

28. = k2(k+1)2 + 4(k+1)3 4 = (k+1)2 [k2 + 4(k+1)] = (k+1)2 [k2 + 4k+4] = (k+1)2 (k+2) … + k3 + (k+1)3 = (k+1)2 (k+2)2 4

29. So we started with the problem:
… + n3 = n2(n+1)2 4

30. So we started with the problem:
… + n3 = n2(n+1)2 4
And we showed:
… + k3 + (k+1)3 = (k+1)2 (k+2)

31. So we started with the problem:
… + n3 = n2(n+1)2 4
And we showed:
… + k3 + (k+1)3 = (k+1)2 (k+2)
So we proved that the statement holds true when we replace n with k + 1 and thus proven.

32. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2

33. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1:

34. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2

35. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2 1 = 1

36. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2 1 = 1 n = 2:

37. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2 1 = 1 n = 2: = 2(3∙2 – 1) 2

38. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2 1 = 1 n = 2: = 2(3∙2 – 1) 2 5 = 5

39. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2 1 = 1 n = 2: = 2(3∙2 – 1) 2 5 = 5 n = 3:

40. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2 1 = 1 n = 2: = 2(3∙2 – 1) 2 5 = 5 n = 3: = 3(3∙3 – 1) 2

41. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2 1 = 1 n = 2: = 2(3∙2 – 1) 2 5 = 5 n = 3: = 3(3∙3 – 1) 2 14 = 12

42. Ex. 2 Find a counterexample to disprove the statement
Ex. 2 Find a counterexample to disprove the statement … + n2 = n(3n – 1) 2 n = 1: 12 = 1(3∙1 – 1) 2 1 = 1 n = 2: = 2(3∙2 – 1) 2 5 = 5 n = 3: = 3(3∙3 – 1) 2 14 = 12, FALSE

43. n = 3 Ex. 2 Find a counterexample to disprove the statement.
… + n2 = n(3n – 1)
n = 1: 12 = 1(3∙1 – 1)
1 = 1
n = 2: = 2(3∙2 – 1)
5 = 5
n = 3: = 3(3∙3 – 1)
14 = 12, FALSE
n = 3