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Mathematical Induction
53
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Page 1: X2 t08 02 induction (2013)

Mathematical Induction

Page 2: X2 t08 02 induction (2013)

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Page 3: X2 t08 02 induction (2013)

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Test: n = 1

Page 4: X2 t08 02 induction (2013)

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Test: n = 1

1

1

1..

2

SHL

Page 5: X2 t08 02 induction (2013)

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Test: n = 1

1

1

1..

2

SHL

1

1

12..

SHR

Page 6: X2 t08 02 induction (2013)

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Test: n = 1

1

1

1..

2

SHL

1

1

12..

SHR

SHRSHL ....

Page 7: X2 t08 02 induction (2013)

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Test: n = 1

1

1

1..

2

SHL

1

1

12..

SHR

SHRSHL ....

kk

knA1

21

3

1

2

11

222

Page 8: X2 t08 02 induction (2013)

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Test: n = 1

1

1

1..

2

SHL

1

1

12..

SHR

SHRSHL ....

kk

knA1

21

3

1

2

11

222

1

12

1

1

3

1

2

11 1

222

kkknP

Page 9: X2 t08 02 induction (2013)

Proof:

22222221

11

3

1

2

11

1

1

3

1

2

11

kkk

Page 10: X2 t08 02 induction (2013)

Proof:

22222221

11

3

1

2

11

1

1

3

1

2

11

kkk

21

112

kk

Page 11: X2 t08 02 induction (2013)

Proof:

22222221

11

3

1

2

11

1

1

3

1

2

11

kkk

21

112

kk

2

2

1

12

kk

kk

Page 12: X2 t08 02 induction (2013)

Proof:

22222221

11

3

1

2

11

1

1

3

1

2

11

kkk

21

112

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

Page 13: X2 t08 02 induction (2013)

Proof:

22222221

11

3

1

2

11

1

1

3

1

2

11

kkk

21

112

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

22

2

1

1

12

kkkk

kk

Page 14: X2 t08 02 induction (2013)

Proof:

22222221

11

3

1

2

11

1

1

3

1

2

11

kkk

21

112

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

22

2

1

1

12

kkkk

kk

21

12

kk

kk

Page 15: X2 t08 02 induction (2013)

Proof:

22222221

11

3

1

2

11

1

1

3

1

2

11

kkk

21

112

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

22

2

1

1

12

kkkk

kk

21

12

kk

kk

1

12

k

Page 16: X2 t08 02 induction (2013)

Proof:

22222221

11

3

1

2

11

1

1

3

1

2

11

kkk

21

112

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

22

2

1

1

12

kkkk

kk

21

12

kk

kk

1

12

k

1

12

1

1

3

1

2

11

222

kk

Page 17: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Page 18: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1

Page 19: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

Page 20: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

2 kaknA

Page 21: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknP

Page 22: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknP

Proof:

Page 23: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknP

Proof:

kk aa 21

Page 24: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknP

Proof:

kk aa 21

22

Page 25: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknP

Proof:

kk aa 21

22

2

4

Page 26: X2 t08 02 induction (2013)

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknP

Proof:

kk aa 21

22

2

4

21 ka

Page 27: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

by; defined are and sequences The nn yxiii

Page 28: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

by; defined are and sequences The nn yxiii

Page 29: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

10

2511

yx

by; defined are and sequences The nn yxiii

Page 30: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

10

2511

yx

10 kk yxknA

by; defined are and sequences The nn yxiii

Page 31: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

10

2511

yx

10 kk yxknA

10 1 11 kk yxknP

by; defined are and sequences The nn yxiii

Page 32: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

10

2511

yx

10 kk yxknA

10 1 11 kk yxknP

Proof:

by; defined are and sequences The nn yxiii

Page 33: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

10

2511

yx

10 kk yxknA

10 1 11 kk yxknP

Proof:

kk

kkkkkk

yx

yxyxyx

2

211

by; defined are and sequences The nn yxiii

Page 34: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

10

2511

yx

10 kk yxknA

10 1 11 kk yxknP

Proof:

kk

kkkkkk

yx

yxyxyx

2

211

10

kk yx

by; defined are and sequences The nn yxiii

Page 35: X2 t08 02 induction (2013)

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

10

2511

yx

10 kk yxknA

10 1 11 kk yxknP

Proof:

kk

kkkkkk

yx

yxyxyx

2

211

10

kk yx

1011 kk yx

by; defined are and sequences The nn yxiii

Page 36: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Page 37: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

Page 38: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

1

.. 1

aSHL

Page 39: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

1

.. 1

aSHL

62.1

2

51..

1

SHR

Page 40: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

1

.. 1

aSHL

62.1

2

51..

1

SHR

SHRSHL ....

Page 41: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

1

.. 1

aSHL

62.1

2

51..

1

SHR

SHRSHL ....

1

.. 2

aSHL

Page 42: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

1

.. 1

aSHL

62.1

2

51..

1

SHR

SHRSHL ....

1

.. 2

aSHL

62.2

2

51..

2

SHR

Page 43: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

1

.. 1

aSHL

62.1

2

51..

1

SHR

SHRSHL ....

1

.. 2

aSHL

62.2

2

51..

2

SHR

SHRSHL ....

Page 44: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

k

k

k

k aaknknA

2

51 &

2

51 &1

1

1

1

.. 1

aSHL

62.1

2

51..

1

SHR

SHRSHL ....

1

.. 2

aSHL

62.2

2

51..

2

SHR

SHRSHL ....

Page 45: X2 t08 02 induction (2013)

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

Test: n = 1 and n =2

k

k

k

k aaknknA

2

51 &

2

51 &1

1

1

1

12

51 1

k

kaknP

1

.. 1

aSHL

62.1

2

51..

1

SHR

SHRSHL ....

1

.. 2

aSHL

62.2

2

51..

2

SHR

SHRSHL ....

Page 46: X2 t08 02 induction (2013)

Proof: 11 kkk aaa

Page 47: X2 t08 02 induction (2013)

Proof: 11 kkk aaa1

2

51

2

51

kk

Page 48: X2 t08 02 induction (2013)

Proof: 11 kkk aaa1

2

51

2

51

kk

211

2

51

2

51

2

51k

Page 49: X2 t08 02 induction (2013)

Proof: 11 kkk aaa1

2

51

2

51

kk

211

2

51

2

51

2

51k

2

1

51

4

51

2

2

51k

Page 50: X2 t08 02 induction (2013)

Proof: 11 kkk aaa1

2

51

2

51

kk

211

2

51

2

51

2

51k

2

1

51

4

51

2

2

51k

2

1

2

1

51

526

2

51

51

4522

2

51

k

k

Page 51: X2 t08 02 induction (2013)

Proof: 11 kkk aaa1

2

51

2

51

kk

211

2

51

2

51

2

51k

2

1

51

4

51

2

2

51k

2

1

2

1

51

526

2

51

51

4522

2

51

k

k

1

2

51

k

Page 52: X2 t08 02 induction (2013)

Proof: 11 kkk aaa1

2

51

2

51

kk

211

2

51

2

51

2

51k

2

1

51

4

51

2

2

51k

2

1

2

1

51

526

2

51

51

4522

2

51

k

k

1

2

51

k

1

12

51

k

ka

Page 53: X2 t08 02 induction (2013)

Proof: 11 kkk aaa1

2

51

2

51

kk

211

2

51

2

51

2

51k

2

1

51

4

51

2

2

51k

2

1

2

1

51

526

2

51

51

4522

2

51

k

k

1

2

51

k

1

12

51

k

ka

Sheets

+

Exercise 10E*