1. Mathematical Induction
Divisibility

2. Review
A function is defined for integers only such that f(n) = 34n -1. Find;
f(1) =
f(2) =
f(3) =
f(4) =

3. Review
A function is defined for integers only such that f(n) = 34n -1. Find;
f(1) =
f(2) =
f(3) =
f(4) =

4. Review
A function is defined for integers only such that f(n) = 34n -1. Find;
f(1) = 80

5. Review
A function is defined for integers only such that f(n) = 34n -1. Find;
f(1) = 80
f(2) = 6560

6. Review
A function is defined for integers only such that f(n) = 34n -1. Find;
f(1) = 80
f(2) = 6560
f(3) =

7. Review
A function is defined for integers only such that f(n) = 34n -1. Find;
f(1) = 80
f(2) = 6560
f(3) =
f(4) =

8. Review
A function is defined for integers only such that f(n) = 34n -1. Find;
f(1) = 80
f(2) = 6560
f(3) =
f(4) =
Now take a guess and make up a theory. What is the highest common factor of your answers……?

9. Review
A function is defined for integers only such that f(n) = 34n -1. Find;
f(1) = 80
f(2) = 6560
f(3) =
f(4) =
Now take a guess and make up a theory. What is the highest common factor of your answers……?

10. Did you get 80?
f(1) = 80 = 80  1
f(2) = 6560 = 80  82
f(3) = = 80  6643
f(4) = = 80  53804
And this is an important idea. If a number is divisible by 80 then we can write it as 80  M where M is a positive integer.

11. Did you get 80?
f(1) = 80 = 80  1
f(2) = 6560 = 80  82
f(3) = = 80  6643
f(4) = = 80  53804
And this is an important idea. If a number is divisible by 80 then we can write it as 80  M where M is a positive integer.

12. So now the theory is that any number of the form 34n -1 where n is a positive integer is divisible by 80.

13. So now the theory is that any number of the form 34n -1 where n is a positive integer is divisible by 80.
So how would we prove it?

14. So now the theory is that any number of the form 34n -1 where n is a positive integer is divisible by 80.
So how would we prove it?
MATHEMATICAL INDUCTION !

15. 1. Prove that any number of the form 34n -1
1. Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.

16. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.

17. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 1 : Prove true for n = 1.
34  1 -1 = 80
= 80  1 which is divisible by 80

18. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 1 : Prove true for n = 1.
34  1 -1 = 80
= 80  1 which is divisible by 80

19. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 2 : Assume true for n = k.
34 k -1 = 80  M
where M is a positive integer

20. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 2 : Assume true for n = k.
34 k -1 = 80  M where M is a positive integer
34 k = 80  M + 1
note that it is very important to make 34 k the subject

21. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 2 : Assume true for n = k.
34k -1 = 80  M where M is a positive integer
34k = 80  M + 1
note that it is very important to make 34k the subject

22. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 2 : Assume true for n = k.
34k -1 = 80  M where M is a positive integer
34k = 80  M + 1
note that it is very important to make 34k the subject

23. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 3 : Prove true for n = k + 1.
RTP that 34(k + 1) -1 is divisible by 80

24. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 3 : Prove true for n = k + 1.
RTP that 34(k + 1) -1 is divisible by 80

25. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 3 : Prove true for n = k + 1.
RTP that 34(k + 1) -1 is divisible by 80

26. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 3 : Prove true for n = k + 1.
RTP that 34(k + 1) -1 is divisible by 80

27. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 3 : Prove true for n = k + 1.
RTP that 34(k + 1) -1 is divisible by 80

28. Prove that any number of the form 34n -1
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 3 : Prove true for n = k + 1.
RTP that 34(k + 1) -1 is divisible by 80

29. Step 3 : Prove true for n = k + 1.
Prove that any number of the form 34n -1 where n is a positive integer is divisible by 80, using mathematical induction.
Step 3 : Prove true for n = k + 1.
RTP that 34(k + 1) -1 is divisible by 80
proved by induction

30. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.

31. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.

32. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.
Step 1 : Prove true for n = 1.

33. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.
Step 1 : Prove true for n = 1.

34. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.
Step 1 : Prove true for n = 1.
= 10
= 2  5 , true for n = 1

35. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.
Step 2 : Assume true for n = k, (k is odd).

36. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.
Step 2 : Assume true for n = k, (k is odd).
3k + 7k = 5 M

37. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.
Step 2 : Assume true for n = k, (k is odd).
3k + 7k = 5 M
7k = 5 M - 3k

38. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.
Step 2 : Assume true for n = k, (k is odd).
3k + 7k = 5 M
7k = 5 M - 3k
note that it is important to make one of them the subject and it is easier to use the bigger number.

39. Prove that any number of the form 3n + 7n
Prove that any number of the form 3n + 7n where n is an odd integer is divisible by 5, using mathematical induction.
Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5

40. Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5

41. Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5

42. Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5

43. Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5

44. Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5

45. Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5

46. Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5
So it is true for n = k + 2, proved by induction

47. Step 3 : Prove true for n = k + 2.
RTP 3k k + 2 is divisible by 5
So it is true for n = k + 2, proved by induction