Mathematical Induction

Mathematical Induction 4for 2 Prove .. 2 nnvge n

Mathematical Induction 4for 2 Prove .. 2 nnvge n

Step 1: Prove the result is true for n = 5

Mathematical Induction 4for 2 Prove .. 2 nnvge n

Step 1: Prove the result is true for n = 5

3225

LHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

Step 1: Prove the result is true for n = 5

3225

LHS

2552

RHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

Step 1: Prove the result is true for n = 5

3225

LHS

2552

RHS

RHSLHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

Step 1: Prove the result is true for n = 5

3225

LHS

Hence the result is true for n = 5

2552

RHS

RHSLHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

Step 1: Prove the result is true for n = 5

3225

LHS

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

22 .. kei k

2552

RHS

RHSLHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

Step 1: Prove the result is true for n = 5

3225

LHS

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

22 .. kei k

Step 3: Prove the result is true for n = k + 1

21 12 :Prove .. kei k

2552

RHS

RHSLHS

Proof:

Proof: 12 k

Proof: 12 k k22

Proof: 12 k k22 22k

Proof: 12 k k22 22k

22 kk

Proof: 12 k k22 22k

22 kk kkk 2

Proof: 12 k k22 22k

22 kk kkk 2

kk 42

Proof: 12 k k22 22k

22 kk kkk 2

kk 42 4k

Proof: 12 k k22 22k

22 kk kkk 2

kk 42 4kkkk 222

Proof: 12 k k22 22k

22 kk kkk 2

kk 42 4kkkk 222

822 kk

Proof: 12 k k22 22k

22 kk kkk 2

kk 42 4kkkk 222

822 kk 4k

Proof: 12 k k22 22k

22 kk kkk 2

kk 42 4kkkk 222

822 kk 4k122 kk

Proof: 12 k k22 22k

22 kk kkk 2

kk 42 4kkkk 222

822 kk 4k122 kk

21 k

Proof: 12 k k22 22k

22 kk kkk 2

kk 42 4kkkk 222

822 kk 4k122 kk

21 k 21 12 kk

Proof: 12 k k22

Hence the result is true for n = k + 1 if it is also true for n = k

22k22 kk

kkk 2

kk 42 4kkkk 222

822 kk 4k122 kk

21 k 21 12 kk

Proof: 12 k k22

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 .

22k22 kk

kkk 2

kk 42 4kkkk 222

822 kk 4k122 kk

21 k 21 12 kk

Proof: 12 k k22

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 .

22k22 kk

kkk 2

kk 42 4kkkk 222

822 kk 4k122 kk

21 k 21 12 kk

Exercise 6N; 6 abc, 8a, 15