12X1 T08 03 binomial theorem

57
Calculating Coefficients without Pascal’s Triangle

Transcript of 12X1 T08 03 binomial theorem

Page 1: 12X1 T08 03 binomial theorem

Calculating Coefficients without Pascal’s Triangle

Page 2: 12X1 T08 03 binomial theorem

Calculating Coefficients without Pascal’s Triangle

!!!

knknCk

n

Page 3: 12X1 T08 03 binomial theorem

Calculating Coefficients without Pascal’s Triangle

Proof

!!!

knknCk

n

Page 4: 12X1 T08 03 binomial theorem

Calculating Coefficients without Pascal’s Triangle

Proof

!!!

knknCk

n

Step 1: Prove true for k = 0

Page 5: 12X1 T08 03 binomial theorem

Calculating Coefficients without Pascal’s Triangle

10

CLHS n

Proof

!!!

knknCk

n

Step 1: Prove true for k = 0

Page 6: 12X1 T08 03 binomial theorem

Calculating Coefficients without Pascal’s Triangle

10

CLHS n

Proof

1!!

!!0!

nn

nnRHS

!!!

knknCk

n

Step 1: Prove true for k = 0

Page 7: 12X1 T08 03 binomial theorem

Calculating Coefficients without Pascal’s Triangle

10

CLHS n

RHSLHS

Proof

1!!

!!0!

nn

nnRHS

!!!

knknCk

n

Step 1: Prove true for k = 0

Page 8: 12X1 T08 03 binomial theorem

Calculating Coefficients without Pascal’s Triangle

10

CLHS n

RHSLHS

Proof

1!!

!!0!

nn

nnRHS

!!!

knknCk

n

Step 1: Prove true for k = 0

Hence the result is true for k = 0

Page 9: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

Page 10: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

Page 11: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

1for trueisresult theProve:3 Step rk

Page 12: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

1for trueisresult theProve:3 Step rk

!1!1!.. 1

rnrnCei r

n

Page 13: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

1for trueisresult theProve:3 Step rk

!1!1!.. 1

rnrnCei r

n

Proof

Page 14: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

1for trueisresult theProve:3 Step rk

!1!1!.. 1

rnrnCei r

n

Proof11

1

rn

rn

rn CCC

Page 15: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

1for trueisresult theProve:3 Step rk

!1!1!.. 1

rnrnCei r

n

Proof11

1

rn

rn

rn CCC

rn

rn

rn CCC

1

11

Page 16: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

1for trueisresult theProve:3 Step rk

!1!1!.. 1

rnrnCei r

n

Proof11

1

rn

rn

rn CCC

rn

rn

rn CCC

1

11

!!

!!!1

!1rnr

nrnr

n

Page 17: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

1for trueisresult theProve:3 Step rk

!1!1!.. 1

rnrnCei r

n

Proof11

1

rn

rn

rn CCC

rn

rn

rn CCC

1

11

!!

!!!1

!1rnr

nrnr

n

!!1

1!!1rnr

rnn

Page 18: 12X1 T08 03 binomial theorem

0integer an is wherefor trueisresult theAssume:2 Step rrk

!!!..

rnrnCei r

n

1for trueisresult theProve:3 Step rk

!1!1!.. 1

rnrnCei r

n

Proof11

1

rn

rn

rn CCC

rn

rn

rn CCC

1

11

!!

!!!1

!1rnr

nrnr

n

!!1

1!!1rnr

rnn

!!1

11!rnr

rnn

Page 19: 12X1 T08 03 binomial theorem

!!1

11!rnr

rnn

Page 20: 12X1 T08 03 binomial theorem

!!1

11!rnr

rnn

!!1

!rnr

rnn

Page 21: 12X1 T08 03 binomial theorem

!!1

11!rnr

rnn

!!1

!rnr

rnn

!1!1!

rnrn

Page 22: 12X1 T08 03 binomial theorem

!!1

11!rnr

rnn

!!1

!rnr

rnn

!1!1!

rnrn

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

Page 23: 12X1 T08 03 binomial theorem

!!1

11!rnr

rnn

!!1

!rnr

rnn

!1!1!

rnrn

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

Step 4: Hence the result is true for all positive integral values of n by induction

Page 24: 12X1 T08 03 binomial theorem

The Binomial Theorem

Page 25: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

Page 26: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

Page 27: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

Page 28: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

NOTE: there are (n + 1) terms

Page 29: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

NOTE: there are (n + 1) terms

This extends to;

n

k

kknk

nn baCba0

Page 30: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

NOTE: there are (n + 1) terms

This extends to;

n

k

kknk

nn baCba0

411Evaluate .. Cge

Page 31: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

NOTE: there are (n + 1) terms

This extends to;

n

k

kknk

nn baCba0

411Evaluate .. Cge

!7!4!11

411 C

Page 32: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

NOTE: there are (n + 1) terms

This extends to;

n

k

kknk

nn baCba0

411Evaluate .. Cge

!7!4!11

411 C

1234891011

Page 33: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

NOTE: there are (n + 1) terms

This extends to;

n

k

kknk

nn baCba0

411Evaluate .. Cge

!7!4!11

411 C

1234891011

Page 34: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

NOTE: there are (n + 1) terms

This extends to;

n

k

kknk

nn baCba0

411Evaluate .. Cge

!7!4!11

411 C

1234891011

3

Page 35: 12X1 T08 03 binomial theorem

The Binomial Theorem n

nnk

knnnnn xCxCxCxCCx 2

2101

n

k

kk

n xC0

integer positive a is and

!!!where n

knknCk

n

NOTE: there are (n + 1) terms

This extends to;

n

k

kknk

nn baCba0

411Evaluate .. Cge

!7!4!11

411 C

1234891011

330

3

Page 36: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

Page 37: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

Page 38: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

Page 39: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

Page 40: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

Page 41: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

Page 42: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

19n

Page 43: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

19n

1135 ofexpansion in the 5th term theFind

baiii

Page 44: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

19n

1135 ofexpansion in the 5th term theFind

baiii

k

kkk b

aCT

35 1111

1

Page 45: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

19n

1135 ofexpansion in the 5th term theFind

baiii

k

kkk b

aCT

35 1111

1

4

74

115

35

baCT

Page 46: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

19n

1135 ofexpansion in the 5th term theFind

baiii

k

kkk b

aCT

35 1111

1

4

74

115

35

baCT

4

7474

11 35b

aC

Page 47: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

19n

1135 ofexpansion in the 5th term theFind

baiii

k

kkk b

aCT

35 1111

1

4

74

115

35

baCT

4

7474

11 35b

aC unsimplified

Page 48: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

19n

1135 ofexpansion in the 5th term theFind

baiii

k

kkk b

aCT

35 1111

1

4

74

115

35

baCT

4

7474

11 35b

aC unsimplified

4

78178125330b

a

Page 49: 12X1 T08 03 binomial theorem

(ii) Find the value of n so that;

85 a) CC nn

knn

kn CC

1358

nn

820

87 b) CCC nn

kn

kn

kn CCC

1

11

19n

1135 ofexpansion in the 5th term theFind

baiii

k

kkk b

aCT

35 1111

1

4

74

115

35

baCT

4

7474

11 35b

aC unsimplified

4

78178125330b

a

4

72088281250b

a

Page 50: 12X1 T08 03 binomial theorem

9

2

213in oft independen termObtain the

xxxiv

Page 51: 12X1 T08 03 binomial theorem

9

2

213in oft independen termObtain the

xxxiv

k

kkk x

xCT

213 929

1

Page 52: 12X1 T08 03 binomial theorem

9

2

213in oft independen termObtain the

xxxiv

k

kkk x

xCT

213 929

1

0 with termmeans oft independen term xx

Page 53: 12X1 T08 03 binomial theorem

9

2

213in oft independen termObtain the

xxxiv

k

kkk x

xCT

213 929

1

0 with termmeans oft independen term xx

0192 xxx kk

Page 54: 12X1 T08 03 binomial theorem

9

2

213in oft independen termObtain the

xxxiv

k

kkk x

xCT

213 929

1

0 with termmeans oft independen term xx

0192 xxx kk

60318

0218

kk

xxx kk

Page 55: 12X1 T08 03 binomial theorem

9

2

213in oft independen termObtain the

xxxiv

k

kkk x

xCT

213 929

1

0 with termmeans oft independen term xx

0192 xxx kk

60318

0218

kk

xxx kk

6

326

97 2

13

x xCT

Page 56: 12X1 T08 03 binomial theorem

9

2

213in oft independen termObtain the

xxxiv

k

kkk x

xCT

213 929

1

0 with termmeans oft independen term xx

0192 xxx kk

60318

0218

kk

xxx kk

6

326

97 2

13

x xCT

6

36

9

23C

Page 57: 12X1 T08 03 binomial theorem

9

2

213in oft independen termObtain the

xxxiv

k

kkk x

xCT

213 929

1

0 with termmeans oft independen term xx

0192 xxx kk

60318

0218

kk

xxx kk

6

326

97 2

13

x xCT

6

36

9

23C

Exercise 5D; 2, 3, 5, 7, 9, 10ac, 12ac, 13,

14, 15ac, 19, 25