Mathematical Induction

Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term

is)

Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term

is)

Step 2: Assume the result is true for n = k, where k is a positive

integer (or another condition that matches the question)

Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term

is)

Step 2: Assume the result is true for n = k, where k is a positive

integer (or another condition that matches the question)

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

Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term

is)

Step 2: Assume the result is true for n = k, where k is a positive

integer (or another condition that matches the question)

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

NOTE: It is important to note in your conclusion that the result is

true for n = k + 1 if it is true for n = k

Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term

is)

Step 2: Assume the result is true for n = k, where k is a positive

integer (or another condition that matches the question)

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

NOTE: It is important to note in your conclusion that the result is

true for n = k + 1 if it is 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

12123

112531 ..

2222 nnnnige

12123

112531 ..

2222 nnnnige

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

12123

112531 ..

2222 nnnnige

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

1

12

LHS

12123

112531 ..

2222 nnnnige

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

1

12

LHS

1

3113

1

121213

1

RHS

12123

112531 ..

2222 nnnnige

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

1

12

LHS

1

3113

1

121213

1

RHS

RHSLHS

12123

112531 ..

2222 nnnnige

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

1

12

LHS

1

3113

1

121213

1

RHS

RHSLHS

Hence the result is true for n = 1

12123

112531 ..

2222 nnnnige

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

1

12

LHS

1

3113

1

121213

1

RHS

RHSLHS

Hence the result is true for n = 1

Step 2: Assume the result is true for n = k, where k is a positive

integer

12123

112531 ..

2222 nnnnige

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

1

12

LHS

1

3113

1

121213

1

RHS

RHSLHS

Hence the result is true for n = 1

Step 2: Assume the result is true for n = k, where k is a positive

integer

12123

112531 ..

2222 kkkkei

12123

112531 ..

2222 nnnnige

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

1

12

LHS

1

3113

1

121213

1

RHS

RHSLHS

Hence the result is true for n = 1

Step 2: Assume the result is true for n = k, where k is a positive

integer

12123

112531 ..

2222 kkkkei

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

12123

112531 ..

2222 nnnnige

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

1

12

LHS

1

3113

1

121213

1

RHS

RHSLHS

Hence the result is true for n = 1

Step 2: Assume the result is true for n = k, where k is a positive

integer

12123

112531 ..

2222 kkkkei

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

321213

112531 :Prove ..

2222 kkkkei

Proof:

Proof:

22222

2222

1212531

12531

kk

k

Proof:

22222

2222

1212531

12531

kk

k

kS

Proof:

22222

2222

1212531

12531

kk

k

kS

1

kT

Proof:

22222

2222

1212531

12531

kk

k

21212123

1 kkkk

kS

1

kT

Proof:

22222

2222

1212531

12531

kk

k

21212123

1 kkkk

kS

1

kT

1212

3

112 kkkk

Proof:

22222

2222

1212531

12531

kk

k

21212123

1 kkkk

kS

1

kT

1212

3

112 kkkk

12312123

1 kkkk

Proof:

22222

2222

1212531

12531

kk

k

21212123

1 kkkk

kS

1

kT

1212

3

112 kkkk

12312123

1 kkkk

352123

1

362123

1

2

2

kkk

kkkk

Proof:

22222

2222

1212531

12531

kk

k

21212123

1 kkkk

kS

1

kT

1212

3

112 kkkk

12312123

1 kkkk

352123

1

362123

1

2

2

kkk

kkkk

321123

1 kkk

Proof:

22222

2222

1212531

12531

kk

k

21212123

1 kkkk

kS

1

kT

1212

3

112 kkkk

12312123

1 kkkk

352123

1

362123

1

2

2

kkk

kkkk

321123

1 kkk

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

n

k n

n

kkii

1 121212

1

n

k n

n

kkii

1 121212

1

121212

1

75

1

53

1

31

1

n

n

nn

n

k n

n

kkii

1 121212

1

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

121212

1

75

1

53

1

31

1

n

n

nn

n

k n

n

kkii

1 121212

1

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

3

1

31

1

LHS

121212

1

75

1

53

1

31

1

n

n

nn

n

k n

n

kkii

1 121212

1

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

3

1

31

1

LHS

3

1

12

1

RHS

121212

1

75

1

53

1

31

1

n

n

nn

n

k n

n

kkii

1 121212

1

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

3

1

31

1

LHS

3

1

12

1

RHS

RHSLHS

121212

1

75

1

53

1

31

1

n

n

nn

n

k n

n

kkii

1 121212

1

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

3

1

31

1

LHS

3

1

12

1

RHS

RHSLHS

Hence the result is true for n = 1

121212

1

75

1

53

1

31

1

n

n

nn

n

k n

n

kkii

1 121212

1

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

3

1

31

1

LHS

3

1

12

1

RHS

RHSLHS

Hence the result is true for n = 1

Step 2: Assume the result is true for n = k, where k is a positive

integer

121212

1

75

1

53

1

31

1

n

n

nn

n

k n

n

kkii

1 121212

1

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

3

1

31

1

LHS

3

1

12

1

RHS

RHSLHS

Hence the result is true for n = 1

Step 2: Assume the result is true for n = k, where k is a positive

integer

121212

1

75

1

53

1

31

1

n

n

nn

121212

1

75

1

53

1

31

1..

k

k

kkei

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

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

32

1

3212

1

75

1

53

1

31

1 :Prove ..

k

k

kkei

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

32

1

3212

1

75

1

53

1

31

1 :Prove ..

k

k

kkei

Proof:

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

32

1

3212

1

75

1

53

1

31

1 :Prove ..

k

k

kkei

Proof:

3212

1

1212

1

75

1

53

1

31

1

3212

1

75

1

53

1

31

1

kkkk

kk

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

32

1

3212

1

75

1

53

1

31

1 :Prove ..

k

k

kkei

Proof:

3212

1

1212

1

75

1

53

1

31

1

3212

1

75

1

53

1

31

1

kkkk

kk

3212

1

12

kkk

k

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

32

1

3212

1

75

1

53

1

31

1 :Prove ..

k

k

kkei

Proof:

3212

1

1212

1

75

1

53

1

31

1

3212

1

75

1

53

1

31

1

kkkk

kk

3212

1

12

kkk

k

3212

132

kk

kk

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

32

1

3212

1

75

1

53

1

31

1 :Prove ..

k

k

kkei

Proof:

3212

1

1212

1

75

1

53

1

31

1

3212

1

75

1

53

1

31

1

kkkk

kk

3212

1

12

kkk

k

3212

132

kk

kk

3212

132 2

kk

kk

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

32

1

3212

1

75

1

53

1

31

1 :Prove ..

k

k

kkei

Proof:

3212

1

1212

1

75

1

53

1

31

1

3212

1

75

1

53

1

31

1

kkkk

kk

3212

1

12

kkk

k

3212

132

kk

kk

3212

132 2

kk

kk

3212

112

kk

kk

32

1

k

k

32

1

k

k

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

32

1

k

k

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

32

1

k

k

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

Exercise 6N; 1 ace etc, 10(polygon), 13