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Transcript of mathematical induction
![Page 1: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/1.jpg)
Mathematical Induction
![Page 2: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/2.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
![Page 3: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/3.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
![Page 4: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/4.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
![Page 5: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/5.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
which is divisible by 3
![Page 6: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/6.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
![Page 7: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/7.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
![Page 8: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/8.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
![Page 9: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/9.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
Step 3: Prove the result is true for n = k + 1
![Page 10: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/10.jpg)
Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
Step 3: Prove the result is true for n = k + 1
integeran is where,3321 :Prove .. QQkkkei
![Page 11: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/11.jpg)
Proof:
![Page 12: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/12.jpg)
Proof: 321 kkk
![Page 13: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/13.jpg)
Proof: 321 kkk
21321 kkkkk
![Page 14: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/14.jpg)
Proof: 321 kkk
21321 kkkkk
2133 kkP
![Page 15: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/15.jpg)
Proof: 321 kkk
21321 kkkkk
2133 kkP
213 kkP
![Page 16: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/16.jpg)
Proof: 321 kkk
21321 kkkkk
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
![Page 17: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/17.jpg)
Proof: 321 kkk
21321 kkkkk
Hence the result is true for n = k + 1 if it is also true for n = k
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
![Page 18: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/18.jpg)
Proof: 321 kkk
21321 kkkkk
Hence the result is true for n = k + 1 if it is also true for n = k
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
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 19: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/19.jpg)
5by divisible is 23 Prove 23 nniv
![Page 20: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/20.jpg)
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
![Page 21: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/21.jpg)
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
![Page 22: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/22.jpg)
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
which is divisible by 5
![Page 23: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/23.jpg)
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
which is divisible by 5
![Page 24: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/24.jpg)
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,523 .. 23 PPei kk
which is divisible by 5
![Page 25: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/25.jpg)
5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,523 .. 23 PPei kk
which is divisible by 5
Step 3: Prove the result is true for n = k + 1
integeran is where,523 :Prove .. 333 QQei kk
![Page 26: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/26.jpg)
Proof:
![Page 27: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/27.jpg)
Proof: 333 23 kk
![Page 28: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/28.jpg)
Proof: 333 23 kk
33 2327 kk
![Page 29: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/29.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP
![Page 30: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/30.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
![Page 31: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/31.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP
![Page 32: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/32.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
![Page 33: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/33.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
![Page 34: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/34.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
![Page 35: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/35.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
Hence the result is true for n = k + 1 if it is also true for n = k
![Page 36: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/36.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
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 37: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042613/5469ecfbaf7959ff128b60b5/html5/thumbnails/37.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
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;
3bdf, 4 ab(i), 9