mathematical induction

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Mathematical Induction

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Page 1: mathematical induction

Mathematical Induction

Page 2: mathematical induction

Mathematical Induction 3by divisible is 21 Prove .. nnniiige

Page 3: mathematical induction

Mathematical Induction 3by divisible is 21 Prove .. nnniiige

Step 1: Prove the result is true for n = 1

Page 4: mathematical induction

Mathematical Induction 3by divisible is 21 Prove .. nnniiige

Step 1: Prove the result is true for n = 1

6

321

Page 5: mathematical induction

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

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

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

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

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

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

Proof:

Page 12: mathematical induction

Proof: 321 kkk

Page 13: mathematical induction

Proof: 321 kkk

21321 kkkkk

Page 14: mathematical induction

Proof: 321 kkk

21321 kkkkk

2133 kkP

Page 15: mathematical induction

Proof: 321 kkk

21321 kkkkk

2133 kkP

213 kkP

Page 16: mathematical induction

Proof: 321 kkk

21321 kkkkk

2133 kkP

213 kkP

integeran is which 21 where,3 kkPQQ

Page 17: mathematical induction

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

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

5by divisible is 23 Prove 23 nniv

Page 20: mathematical induction

5by divisible is 23 Prove 23 nniv

Step 1: Prove the result is true for n = 1

Page 21: mathematical induction

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

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

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

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

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

Proof:

Page 27: mathematical induction

Proof: 333 23 kk

Page 28: mathematical induction

Proof: 333 23 kk

33 2327 kk

Page 29: mathematical induction

Proof: 333 23 kk

33 2327 kk

32 22527 kkP

Page 30: mathematical induction

Proof: 333 23 kk

33 2327 kk

32 22527 kkP32 2227135 kkP

Page 31: mathematical induction

Proof: 333 23 kk

33 2327 kk

32 22527 kkP32 2227135 kkP

22 22227135 kkP

Page 32: mathematical induction

Proof: 333 23 kk

33 2327 kk

32 22527 kkP32 2227135 kkP

22 22227135 kkP2225135 kP

Page 33: mathematical induction

Proof: 333 23 kk

33 2327 kk

32 22527 kkP32 2227135 kkP

22 22227135 kkP2225135 kP

225275 kP

Page 34: mathematical induction

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

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

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

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