11x1 t14 10 mathematical induction 3 (2011)

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

32

25

LHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

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

32

25

LHS

25

52

RHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

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

32

25

LHS

25

52

RHS

RHSLHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

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

32

25

LHS

Hence the result is true for n = 5

25

52

RHS

RHSLHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

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

32

25

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

25

52

RHS

RHSLHS

Mathematical Induction 4for 2 Prove .. 2 nnvge n

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

32

25

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

25

52

RHS

RHSLHS

Proof:

Proof: 12 k

Proof: 12 k k22

Proof: 12 k k22

22k

Proof: 12 k k22

22k22 kk

Proof: 12 k k22

22k22 kk

kkk 2

Proof: 12 k k22

22k22 kk

kkk 2

kk 42

Proof: 12 k k22

22k22 kk

kkk 2

kk 42 4k

Proof: 12 k k22

22k22 kk

kkk 2

kk 42 4k

kkk 222

Proof: 12 k k22

22k22 kk

kkk 2

kk 42 4k

kkk 222

822 kk

Proof: 12 k k22

22k22 kk

kkk 2

kk 42 4k

kkk 222

822 kk 4k

Proof: 12 k k22

22k22 kk

kkk 2

kk 42 4k

kkk 222

822 kk 4k

122 kk

Proof: 12 k k22

22k22 kk

kkk 2

kk 42 4k

kkk 222

822 kk 4k

122 kk

21 k

Proof: 12 k k22

22k22 kk

kkk 2

kk 42 4k

kkk 222

822 kk 4k

122 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 4k

kkk 222

822 kk 4k

122 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 4k

kkk 222

822 kk 4k

122 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 4k

kkk 222

822 kk 4k

122 kk

21 k

21 12 kk

Exercise 6N;

6 abc, 8a, 15