1. By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org
Math Induction
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: March 30, 2011

2. Before we begin…… If Pk = 1 + 5 + 9 + . . . + (4k-3) find Pk+1.
Pk+1 = (4k-3) + (4(k+1) – 3)
Pk+1 = (4k-3) + (4k+1)

3. Before we begin…… If Pk = k2(k+1)2 find Pk+1.
Pk+1 = (k+1)2(k+1+1)2 = (k+1)2(k+2)2

4. The Principle of Math Induction
Let Pn be a statement involving the positive integer n. If
1. P1 is true
the truth of Pk implies the truth of Pk+1 for every positive k
then Pn must be true for all positive integers n.
To apply the Principle of Math Induction, you need to be able to determine the statement Pk+1 for a given statement Pk. To determine Pk+1, substitute k+1 for k in the statement Pk.
Direct Quote: Precalculus with Limits – A Graphin Approach by Larson, Hostetler & Edwards © 2005 by Houghton Mifflin Company

5. Induction – A three step process
Step 1: Show P1 is true
APPROACH
THE
Step 2: Assume Pk is true
Step 3: Prove Pk+1 is true
A “Bivinism”

6. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

7. Prove Pk+1:
Q.E.D.

8. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

9. Prove Pk+1:
Q.E.D.

10. Prove:
Show P1:
Assume Pk:
Prove Pk+1:
Q.E.D.

11. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

12. Prove Pk+1:
Q.E.D.

13. Worksheet
Exercise # 1

14. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

15. Prove Pk+1:
Q.E.D.

16. Worksheet
Exercise # 2

17. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

18. Prove Pk+1:
Q.E.D.

19. Worksheet
Exercise # 3

20. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

21. Prove Pk+1:
Q.E.D.

22. Worksheet
Exercise # 4

23. Prove:
Show P1:
Assume Pk:
Prove Pk+1:
Q.E.D.

24. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

25. Worksheet
Exercise # 5

26. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

27. Prove Pk+1:
Q.E.D.

28. Worksheet
Exercise # 6

29. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

30. Prove Pk+1:
Q.E.D.

31. Worksheet
Exercise # 7

32. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

33. Prove Pk+1:
Q.E.D.

34. Worksheet
Exercise # 8

35. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

36. Prove Pk+1:
Q.E.D.

37. Worksheet
Exercise # 9

38. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

39. We need a common denominator to add the fractions.
Prove Pk+1:
We need a common denominator to add the fractions.

40. Prove Pk+1:
We see that a factor of (k+1) is in the denominator on the left that is not on the right. Maybe the numerator has a (k+1) factor that will cancel with it. We will use synthetic division to check. 3 14 19 8 -1 -3
-11 -8 11
Q.E.D.

41. Worksheet
Exercise # 10

42. Prove:
Show P1:
Assume Pk:
Prove Pk+1:

43. Prove Pk+1:
Q.E.D.