11x1 t14 09 mathematical induction 2 (2011)

Mathematical Induction

Mathematical Induction 3by divisible is 21 Prove .. nnniiige

Mathematical Induction 3by divisible is 21 Prove .. nnniiige

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

Mathematical Induction 3by divisible is 21 Prove .. nnniiige

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

6

321

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

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

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

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

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

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

Proof:

Proof: 321 kkk

Proof: 321 kkk

21321 kkkkk

Proof: 321 kkk

21321 kkkkk

2133 kkP

Proof: 321 kkk

21321 kkkkk

2133 kkP

213 kkP

Proof: 321 kkk

21321 kkkkk

2133 kkP

213 kkP

integeran is which 21 where,3 kkPQQ

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

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.

5by divisible is 23 Prove 23 nniv

5by divisible is 23 Prove 23 nniv

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

5by divisible is 23 Prove 23 nniv

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

35

827

23 33

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

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

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

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

Proof:

Proof: 333 23 kk

Proof: 333 23 kk

33 2327 kk

Proof: 333 23 kk

33 2327 kk

32 22527 kkP

Proof: 333 23 kk

33 2327 kk

32 22527 kkP32 2227135 kkP

Proof: 333 23 kk

33 2327 kk

32 22527 kkP32 2227135 kkP

22 22227135 kkP

Proof: 333 23 kk

33 2327 kk

32 22527 kkP32 2227135 kkP

22 22227135 kkP2225135 kP

Proof: 333 23 kk

33 2327 kk

32 22527 kkP32 2227135 kkP

22 22227135 kkP2225135 kP

225275 kP

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

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

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.

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