Download - 11X1 T10 08 mathematical induction 1
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Mathematical Induction
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Mathematical InductionStep 1: Prove the result is true for n = 1 (or whatever the first term
is)
![Page 3: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/3.jpg)
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)
![Page 4: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/4.jpg)
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
![Page 5: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/5.jpg)
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
![Page 6: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/6.jpg)
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
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12123112531 .. 2222 nnnnige
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12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
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12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
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12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
RHS
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12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
RHS
RHSLHS
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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
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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
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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
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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
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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
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Proof:
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Proof: 22222
2222
1212531
12531
kk
k
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Proof: 22222
2222
1212531
12531
kk
k
kS
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Proof: 22222
2222
1212531
12531
kk
k
kS
1
kT
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Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
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Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
![Page 23: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/23.jpg)
Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
123121231
kkkk
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Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
123121231
kkkk
3521231
3621231
2
2
kkk
kkkk
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Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
123121231
kkkk
3521231
3621231
2
2
kkk
kkkk
3211231
kkk
![Page 26: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/26.jpg)
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
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Step 4: Since the result is true for n = 1, then the result is true for all positive integral values of n by induction
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n
k nn
kkii
1 1212121
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n
k nn
kkii
1 1212121
1212121
751
531
311
nn
nn
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n
k nn
kkii
1 1212121
Step 1: Prove the result is true for n = 1 121212
175
153
131
1
n
nnn
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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
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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
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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
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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
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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
![Page 36: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/36.jpg)
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
![Page 37: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/37.jpg)
Step 3: Prove the result is true for n = k + 1
![Page 38: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/38.jpg)
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
![Page 39: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/39.jpg)
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
Proof:
![Page 40: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/40.jpg)
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
![Page 41: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/41.jpg)
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
![Page 42: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/42.jpg)
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
![Page 43: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/43.jpg)
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
![Page 44: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/44.jpg)
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
![Page 45: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/45.jpg)
32
1
kk
![Page 46: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/46.jpg)
32
1
kk
Hence the result is true for n = k + 1 if it is also true for n = k
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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
![Page 48: 11X1 T10 08 mathematical induction 1](https://reader033.vdocument.in/reader033/viewer/2022052906/558e0df91a28ab72318b4721/html5/thumbnails/48.jpg)
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