5. Mathematical Induction
e.g .v Prove 2 n n 2 for n 4
Step 1: Prove the result is true for n = 5
LHS 25 RHS 52
32 25
6. Mathematical Induction
e.g .v Prove 2 n n 2 for n 4
Step 1: Prove the result is true for n = 5
LHS 25 RHS 52
32 25
LHS RHS
7. Mathematical Induction
e.g .v Prove 2 n n 2 for n 4
Step 1: Prove the result is true for n = 5
LHS 25 RHS 52
32 25
LHS RHS
Hence the result is true for n = 5
8. Mathematical Induction
e.g .v Prove 2 n n 2 for n 4
Step 1: Prove the result is true for n = 5
LHS 25 RHS 52
32 25
LHS RHS
Hence the result is true for n = 5
Step 2: Assume the result is true for n = k, where k is a positive
integer > 4
i.e. 2 k k 2
9. Mathematical Induction
e.g .v Prove 2 n n 2 for n 4
Step 1: Prove the result is true for n = 5
LHS 25 RHS 52
32 25
LHS RHS
Hence the result is true for n = 5
Step 2: Assume the result is true for n = k, where k is a positive
integer > 4
i.e. 2 k k 2
Step 3: Prove the result is true for n = k + 1
i.e. Prove : 2 k 1 k 1
2
15. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
16. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k
17. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
18. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
k 2 2k 2k
19. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
k 2 2k 2k
k 2 2k 8
20. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
k 2 2k 2k
k 2 2k 8 k 4
21. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
k 2 2k 2k
k 2 2k 8 k 4
k 2 2k 1
22. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
k 2 2k 2k
k 2 2k 8 k 4
k 2 2k 1
k 1
2
23. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
k 2 2k 2k
k 2 2k 8 k 4
k 2 2k 1
k 1
2
2 k 1 k 1
2
24. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
k 2 2k 2k
k 2 2k 8 k 4
k 2 2k 1
k 1
2
2 k 1 k 1
2
Hence the result is true for n = k + 1 if it is also true for n = k
25. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4
k 2 2k 2k
k 2 2k 8 k 4
k 2 2k 1
k 1
2
2 k 1 k 1
2
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 = 5, then the result is true for
all positive integral values of n > 4 by induction .
26. Proof:
2 k 1 2 2 k
2k 2
k2 k2
k2 k k
k 2 4k k 4 Exercise 6N;
k 2 2k 2k 6 abc, 8a, 15
k 2 2k 8 k 4
k 2 2k 1
k 1
2
2 k 1 k 1
2
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 = 5, then the result is true for
all positive integral values of n > 4 by induction .