11X1 T14 10 mathematical induction 3 (2011)
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Transcript of 11X1 T14 10 mathematical induction 3 (2011)
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Mathematical Induction
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
![Page 4: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/4.jpg)
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
![Page 5: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/5.jpg)
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
25
52
RHS
![Page 6: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/6.jpg)
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
25
52
RHS
RHSLHS
![Page 7: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/7.jpg)
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
Hence the result is true for n = 5
25
52
RHS
RHSLHS
![Page 8: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/8.jpg)
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
Hence the result is true for n = 5
Step 2: Assume the result is true for n = k, where k is a positive
integer > 4 22 .. kei k
25
52
RHS
RHSLHS
![Page 9: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/9.jpg)
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
Hence the result is true for n = 5
Step 2: Assume the result is true for n = k, where k is a positive
integer > 4 22 .. kei k
Step 3: Prove the result is true for n = k + 1
21 12 :Prove .. kei k
25
52
RHS
RHSLHS
![Page 10: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/10.jpg)
Proof:
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Proof: 12 k
![Page 12: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/12.jpg)
Proof: 12 k k22
![Page 13: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/13.jpg)
Proof: 12 k k22
22k
![Page 14: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/14.jpg)
Proof: 12 k k22
22k22 kk
![Page 15: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/15.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
![Page 16: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/16.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
kk 42
![Page 17: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/17.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
![Page 18: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/18.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
![Page 19: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/19.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk
![Page 20: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/20.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
![Page 21: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/21.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
![Page 22: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/22.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
![Page 23: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/23.jpg)
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
21 12 kk
![Page 24: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/24.jpg)
Proof: 12 k k22
Hence the result is true for n = k + 1 if it is also true for n = k
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
21 12 kk
![Page 25: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/25.jpg)
Proof: 12 k k22
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 = 5, then the result is true for
all positive integral values of n > 4 by induction .
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
21 12 kk
![Page 26: 11X1 T14 10 mathematical induction 3 (2011)](https://reader034.fdocuments.us/reader034/viewer/2022042700/558e0e251a28ab6a318b4772/html5/thumbnails/26.jpg)
Proof: 12 k k22
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 = 5, then the result is true for
all positive integral values of n > 4 by induction .
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
21 12 kk
Exercise 6N;
6 abc, 8a, 15