12 x1 t08 03 binomial theorem (12)
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Transcript of 12 x1 t08 03 binomial theorem (12)
![Page 1: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/1.jpg)
Calculating Coefficients without Pascal’s Triangle
![Page 2: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/2.jpg)
Calculating Coefficients without Pascal’s Triangle
!!!
knknCk
n
![Page 3: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/3.jpg)
Calculating Coefficients without Pascal’s Triangle
Proof
!!!
knknCk
n
![Page 4: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/4.jpg)
Calculating Coefficients without Pascal’s Triangle
Proof
!!!
knknCk
n
Step 1: Prove true for k = 0
![Page 5: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/5.jpg)
Calculating Coefficients without Pascal’s Triangle
10
CLHS n
Proof
!!!
knknCk
n
Step 1: Prove true for k = 0
![Page 6: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/6.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/7.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/8.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/9.jpg)
0integer an is wherefor trueisresult theAssume:2 Step rrk
![Page 10: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/10.jpg)
0integer an is wherefor trueisresult theAssume:2 Step rrk
!!!..
rnrnCei r
n
![Page 11: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/11.jpg)
0integer an is wherefor trueisresult theAssume:2 Step rrk
!!!..
rnrnCei r
n
1for trueisresult theProve:3 Step rk
![Page 12: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/12.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/13.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/14.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/15.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/16.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/17.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/18.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/19.jpg)
!!1
11!rnr
rnn
![Page 20: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/20.jpg)
!!1
11!rnr
rnn
!!1
!rnr
rnn
![Page 21: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/21.jpg)
!!1
11!rnr
rnn
!!1
!rnr
rnn
!1!1!
rnrn
![Page 22: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/22.jpg)
!!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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/23.jpg)
!!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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/24.jpg)
The Binomial Theorem
![Page 25: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/25.jpg)
The Binomial Theorem n
nnk
knnnnn xCxCxCxCCx 2
2101
![Page 26: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/26.jpg)
The Binomial Theorem n
nnk
knnnnn xCxCxCxCCx 2
2101
n
k
kk
n xC0
![Page 27: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/27.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/28.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/29.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/30.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/31.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/32.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/33.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/34.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/35.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/36.jpg)
(ii) Find the value of n so that;
![Page 37: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/37.jpg)
(ii) Find the value of n so that;
85 a) CC nn
![Page 38: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/38.jpg)
(ii) Find the value of n so that;
85 a) CC nn
knn
kn CC
![Page 39: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/39.jpg)
(ii) Find the value of n so that;
85 a) CC nn
knn
kn CC
1358
nn
![Page 40: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/40.jpg)
(ii) Find the value of n so that;
85 a) CC nn
knn
kn CC
1358
nn
820
87 b) CCC nn
![Page 41: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/41.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/42.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/43.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/44.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/45.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/46.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/47.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/48.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/49.jpg)
(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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/50.jpg)
9
2
213in oft independen termObtain the
xxxiv
![Page 51: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/51.jpg)
9
2
213in oft independen termObtain the
xxxiv
k
kkk x
xCT
213 929
1
![Page 52: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/52.jpg)
9
2
213in oft independen termObtain the
xxxiv
k
kkk x
xCT
213 929
1
0 with termmeans oft independen term xx
![Page 53: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/53.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/54.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/55.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/56.jpg)
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: 12 x1 t08 03 binomial theorem (12)](https://reader031.fdocuments.net/reader031/viewer/2022020116/5597df171a28ab6e388b472a/html5/thumbnails/57.jpg)
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