x2 t08 02 induction

Mathematical Induction

Mathematical Induction

nni 121

31

211 Prove e.g. 222

Mathematical Induction

nni 121

31

211 Prove e.g. 222

Test: n = 1

Mathematical Induction

nni 121

31

211 Prove e.g. 222

Test: n = 1

111.. 2

SHL

Mathematical Induction

nni 121

31

211 Prove e.g. 222

Test: n = 1

111.. 2

SHL

1112..

SHR

Mathematical Induction

nni 121

31

211 Prove e.g. 222

Test: n = 1

111.. 2

SHL

1112..

SHR

SHRSHL ....

Mathematical Induction

nni 121

31

211 Prove e.g. 222

Test: n = 1

111.. 2

SHL

1112..

SHR

SHRSHL ....

kk

knA 12131

211 222

Mathematical Induction

nni 121

31

211 Prove e.g. 222

Test: n = 1

111.. 2

SHL

1112..

SHR

SHRSHL ....

kk

knA 12131

211 222

1

121

131

211 1 222

kk

knP

Proof:

2222222 111

31

211

11

31

211

kkk

Proof:

2222222 111

31

211

11

31

211

kkk

21112

kk

Proof:

2222222 111

31

211

11

31

211

kkk

21112

kk

2

2

112

kk

kk

Proof:

2222222 111

31

211

11

31

211

kkk

21112

kk

2

2

112

kk

kk

22

112

kk

kk

Proof:

2222222 111

31

211

11

31

211

kkk

21112

kk

2

2

112

kk

kk

22

112

kk

kk

22

2

11

12

kkkk

kk

Proof:

2222222 111

31

211

11

31

211

kkk

21112

kk

2

2

112

kk

kk

22

112

kk

kk

22

2

11

12

kkkk

kk

21

12

kkkk

Proof:

2222222 111

31

211

11

31

211

kkk

21112

kk

2

2

112

kk

kk

22

112

kk

kk

22

2

11

12

kkkk

kk

21

12

kkkk

112

k

Proof:

2222222 111

31

211

11

31

211

kkk

21112

kk

2

2

112

kk

kk

22

112

kk

kk

22

2

11

12

kkkk

kk

21

12

kkkk

112

k

112

11

31

211 222

kk

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1 221 a

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1 221 a

2 kaknA

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknP

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknPProof:

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknPProof:

kk aa 21

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknPProof:

kk aa 21

22

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknPProof:

kk aa 21

22

24

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2that Show nan

Test: n = 1 221 a

2 kaknA

2 1 1 kaknPProof:

kk aa 21

22

24

21 ka

nn

nnn

nnn yx

yxyyxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

by;definedareandsequencesThe nn yxiii

nn

nnn

nnn yx

yxyyxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1

by;definedareandsequencesThe nn yxiii

nn

nnn

nnn yx

yxyyxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1 10

2511

yx

by;definedareandsequencesThe nn yxiii

nn

nnn

nnn yx

yxyyxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1 10

2511

yx

10 kk yxknA

by;definedareandsequencesThe nn yxiii

nn

nnn

nnn yx

yxyyxxyx

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;definedareandsequencesThe nn yxiii

nn

nnn

nnn yx

yxyyxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1 10

2511

yx

10 kk yxknA

10 1 11 kk yxknPProof:

by;definedareandsequencesThe nn yxiii

nn

nnn

nnn yx

yxyyxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1 10

2511

yx

10 kk yxknA

10 1 11 kk yxknPProof:

kk

kkkkkk yx

yxyxyx 2211

by;definedareandsequencesThe nn yxiii

nn

nnn

nnn yx

yxyyxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1 10

2511

yx

10 kk yxknA

10 1 11 kk yxknPProof:

kk

kkkkkk yx

yxyxyx 2211

10 kk yx

by;definedareandsequencesThe nn yxiii

nn

nnn

nnn yx

yxyyxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

Test: n = 1 10

2511

yx

10 kk yxknA

10 1 11 kk yxknPProof:

kk

kkkkkk yx

yxyxyx 2211

10 kk yx

1011 kk yx

by;definedareandsequencesThe nn yxiii

1for 1 1121 naaaaa nnn

(iv) The Fibonacci sequence is defined by;

1for 2

51 that Prove

na

n

n

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

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

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

62.12

51..1

SHR

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

62.12

51..1

SHR

SHRSHL ....

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

62.12

51..1

SHR

SHRSHL ....

1.. 2

aSHL

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

62.12

51..1

SHR

SHRSHL ....

1.. 2

aSHL

62.22

51..2

SHR

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

62.12

51..1

SHR

SHRSHL ....

1.. 2

aSHL

62.22

51..2

SHR

SHRSHL ....

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

251 &

251 &1

1

1

1.. 1

aSHL

62.12

51..1

SHR

SHRSHL ....

1.. 2

aSHL

62.22

51..2

SHR

SHRSHL ....

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

251 &

251 &1

1

1

1

1 251 1

k

kaknP

1.. 1

aSHL

62.12

51..1

SHR

SHRSHL ....

1.. 2

aSHL

62.22

51..2

SHR

SHRSHL ....

Proof: 11 kkk aaa

Proof: 11 kkk aaa1

251

251

kk

Proof: 11 kkk aaa1

251

251

kk

211

251

251

251

k

Proof: 11 kkk aaa1

251

251

kk

211

251

251

251

k

2

1

514

512

251

k

Proof: 11 kkk aaa1

251

251

kk

211

251

251

251

k

2

1

514

512

251

k

2

1

2

1

51526

251

514522

251

k

k

Proof: 11 kkk aaa1

251

251

kk

211

251

251

251

k

2

1

514

512

251

k

2

1

2

1

51526

251

514522

251

k

k

1

251

k

Proof: 11 kkk aaa1

251

251

kk

211

251

251

251

k

2

1

514

512

251

k

2

1

2

1

51526

251

514522

251

k

k

1

251

k

1

1 251

k

ka

Proof: 11 kkk aaa1

251

251

kk

211

251

251

251

k

2

1

514

512

251

k

2

1

2

1

51526

251

514522

251

k

k

1

251

k

1

1 251

k

ka

Sheets

+

Exercise 10E*