3. Mathematical Induction
e.g . iii Prove n n 1 n 2 is divisible by 3
Step 1: Prove the result is true for n = 1
4. Mathematical Induction
e.g . iii Prove n n 1 n 2 is divisible by 3
Step 1: Prove the result is true for n = 1
123
6
5. Mathematical Induction
e.g . iii Prove n n 1 n 2 is divisible by 3
Step 1: Prove the result is true for n = 1
123
6 which is divisible by 3
6. Mathematical Induction
e.g . iii Prove n n 1 n 2 is divisible by 3
Step 1: Prove the result is true for n = 1
123
6 which is divisible by 3
Hence the result is true for n = 1
7. Mathematical Induction
e.g . iii Prove n n 1 n 2 is divisible by 3
Step 1: Prove the result is true for n = 1
123
6 which is divisible by 3
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
8. Mathematical Induction
e.g . iii Prove n n 1 n 2 is divisible by 3
Step 1: Prove the result is true for n = 1
123
6 which is divisible by 3
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
i.e. k k 1k 2 3P, where P is an integer
9. Mathematical Induction
e.g . iii Prove n n 1 n 2 is divisible by 3
Step 1: Prove the result is true for n = 1
123
6 which is divisible by 3
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
i.e. k k 1k 2 3P, where P is an integer
Step 3: Prove the result is true for n = k + 1
10. Mathematical Induction
e.g . iii Prove n n 1 n 2 is divisible by 3
Step 1: Prove the result is true for n = 1
123
6 which is divisible by 3
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
i.e. k k 1k 2 3P, where P is an integer
Step 3: Prove the result is true for n = k + 1
i.e. Prove : k 1k 2 k 3 3Q, where Q is an integer
16. Proof:
k 1k 2k 3
k k 1k 2 3k 1k 2
3P 3k 1k 2
3P k 1k 2
3Q, where Q P k 1k 2 which is an integer
17. Proof:
k 1k 2k 3
k k 1k 2 3k 1k 2
3P 3k 1k 2
3P k 1k 2
3Q, where Q P k 1k 2 which is an integer
Hence the result is true for n = k + 1 if it is also true for n = k
18. Proof:
k 1k 2k 3
k k 1k 2 3k 1k 2
3P 3k 1k 2
3P k 1k 2
3Q, where Q P k 1k 2 which is an integer
Hence the result is true for n = k + 1 if it is also true for n = k
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction.
20. iv Prove 33n 2 n2 is divisible by 5
Step 1: Prove the result is true for n = 1
21. iv Prove 33n 2 n2 is divisible by 5
Step 1: Prove the result is true for n = 1
33 23
27 8
35
22. iv Prove 33n 2 n2 is divisible by 5
Step 1: Prove the result is true for n = 1
33 23
27 8
35 which is divisible by 5
23. iv Prove 33n 2 n2 is divisible by 5
Step 1: Prove the result is true for n = 1
33 23
27 8
35 which is divisible by 5
Hence the result is true for n = 1
24. iv Prove 33n 2 n2 is divisible by 5
Step 1: Prove the result is true for n = 1
33 23
27 8
35 which is divisible by 5
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
i.e. 33k 2 k 2 5 P, where P is an integer
25. iv Prove 33n 2 n2 is divisible by 5
Step 1: Prove the result is true for n = 1
33 23
27 8
35 which is divisible by 5
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
i.e. 33k 2 k 2 5 P, where P is an integer
Step 3: Prove the result is true for n = k + 1
i.e. Prove : 33k 3 2 k 3 5Q, where Q is an integer
29. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
30. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
135 P 27 2 k 2 2 k 3
31. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
135 P 27 2 k 2 2 k 3
135 P 27 2 k 2 2 2 k 2
32. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
135 P 27 2 k 2 2 k 3
135 P 27 2 k 2 2 2 k 2
135 P 25 2 k 2
33. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
135 P 27 2 k 2 2 k 3
135 P 27 2 k 2 2 2 k 2
135 P 25 2 k 2
527 P 5 2 k 2
34. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
135 P 27 2 k 2 2 k 3
135 P 27 2 k 2 2 2 k 2
135 P 25 2 k 2
527 P 5 2 k 2
5Q, where Q 27 P 5 2 k 2 which is an integer
35. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
135 P 27 2 k 2 2 k 3
135 P 27 2 k 2 2 2 k 2
135 P 25 2 k 2
527 P 5 2 k 2
5Q, where Q 27 P 5 2 k 2 which is an integer
Hence the result is true for n = k + 1 if it is also true for n = k
36. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
135 P 27 2 k 2 2 k 3
135 P 27 2 k 2 2 2 k 2
135 P 25 2 k 2
527 P 5 2 k 2
5Q, where Q 27 P 5 2 k 2 which is an integer
Hence the result is true for n = k + 1 if it is also true for n = k
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction.
37. Proof: 33k 3 2 k 3
27 33k 2 k 3
275 P 2 k 2 2 k 3
135 P 27 2 k 2 2 k 3 Exercise 6N;
3bdf, 4 ab(i), 9
135 P 27 2 k 2 2 2 k 2
135 P 25 2 k 2
527 P 5 2 k 2
5Q, where Q 27 P 5 2 k 2 which is an integer
Hence the result is true for n = k + 1 if it is also true for n = k
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction.