11x1 t10 08 mathematical induction 1

Mathematical Induction

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

is)

Mathematical InductionStep 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 InductionStep 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 InductionStep 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 + 1NOTE: 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 InductionStep 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 + 1NOTE: It is important to note in your conclusion that the result is

true for n = k + 1 if it is true for n = kStep 4: Since the result is true for n = 1, then the result is true for

all positive integral values of n by induction

12123112531 .. 2222 nnnnige

12123112531 .. 2222 nnnnige

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

12123112531 .. 2222 nnnnige

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

112

LHS

12123112531 .. 2222 nnnnige

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

112

LHS

1

31131

1212131

RHS

12123112531 .. 2222 nnnnige

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

112

LHS

1

31131

1212131

RHS

RHSLHS

12123112531 .. 2222 nnnnige

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

112

LHS

1

31131

1212131

RHS

RHSLHS Hence the result is true for n = 1

12123112531 .. 2222 nnnnige

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

112

LHS

1

31131

1212131

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

12123112531 .. 2222 nnnnige

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

112

LHS

1

31131

1212131

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

12123112531 .. 2222 kkkkei

12123112531 .. 2222 nnnnige

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

112

LHS

1

31131

1212131

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

12123112531 .. 2222 kkkkei

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

12123112531 .. 2222 nnnnige

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

112

LHS

1

31131

1212131

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

12123112531 .. 2222 kkkkei

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

321213112531 :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

212121231

kkkk

kS

1

kT

Proof: 22222

2222

1212531

12531

kk

k

212121231

kkkk

kS

1

kT

12123112 kkkk

Proof: 22222

2222

1212531

12531

kk

k

212121231

kkkk

kS

1

kT

12123112 kkkk

123121231

kkkk

Proof: 22222

2222

1212531

12531

kk

k

212121231

kkkk

kS

1

kT

12123112 kkkk

123121231

kkkk

3521231

3621231

2

2

kkk

kkkk

Proof: 22222

2222

1212531

12531

kk

k

212121231

kkkk

kS

1

kT

12123112 kkkk

123121231

kkkk

3521231

3621231

2

2

kkk

kkkk

3211231

kkk

Proof: 22222

2222

1212531

12531

kk

k

212121231

kkkk

kS

1

kT

12123112 kkkk

123121231

kkkk

3521231

3621231

2

2

kkk

kkkk

3211231

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 nn

kkii

1 1212121

n

k nn

kkii

1 1212121

1212121

751

531

311

nn

nn

n

k nn

kkii

1 1212121

Step 1: Prove the result is true for n = 1 121212

175

153

131

1

n

nnn

n

k nn

kkii

1 1212121

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

31

311

LHS

1212121

751

531

311

nn

nn

n

k nn

kkii

1 1212121

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

31

311

LHS

31

121

RHS

1212121

751

531

311

nn

nn

n

k nn

kkii

1 1212121

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

31

311

LHS

31

121

RHS

RHSLHS

1212121

751

531

311

nn

nn

n

k nn

kkii

1 1212121

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

31

311

LHS

31

121

RHS

RHSLHS Hence the result is true for n = 1

1212121

751

531

311

nn

nn

n

k nn

kkii

1 1212121

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

31

311

LHS

31

121

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

1212121

751

531

311

nn

nn

n

k nn

kkii

1 1212121

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

31

311

LHS

31

121

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

1212121

751

531

311

nn

nn

1212121

751

531

311..

kk

kkei

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

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

321

32121

751

531

311 :Prove ..

k

kkk

ei

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

321

32121

751

531

311 :Prove ..

k

kkk

ei

Proof:

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

321

32121

751

531

311 :Prove ..

k

kkk

ei

Proof:

32121

12121

751

531

311

32121

751

531

311

kkkk

kk

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

321

32121

751

531

311 :Prove ..

k

kkk

ei

Proof:

32121

12121

751

531

311

32121

751

531

311

kkkk

kk

32121

12

kkkk

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

321

32121

751

531

311 :Prove ..

k

kkk

ei

Proof:

32121

12121

751

531

311

32121

751

531

311

kkkk

kk

32121

12

kkkk

3212

132

kk

kk

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

321

32121

751

531

311 :Prove ..

k

kkk

ei

Proof:

32121

12121

751

531

311

32121

751

531

311

kkkk

kk

32121

12

kkkk

3212

132

kk

kk

3212132 2

kkkk

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

321

32121

751

531

311 :Prove ..

k

kkk

ei

Proof:

32121

12121

751

531

311

32121

751

531

311

kkkk

kk

32121

12

kkkk

3212

132

kk

kk

3212132 2

kkkk

3212

112

kkkk

32

1

kk

32

1

kk

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

32

1

kk

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

kk

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