X2 harder induction

Mathematical Induction

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Test: n = 1

Mathematical Induction

nni

12

1

3

1

2

11 Prove e.g.

222

Test: n = 1

1

1

1..

2

SHL

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

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 ....

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

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

Proof:

2222222 111

3

1

2

11

1

1

3

1

2

11

kkk

Proof:

2222222 111

3

1

2

11

1

1

3

1

2

11

kkk

2111

2

kk

Proof:

2222222 111

3

1

2

11

1

1

3

1

2

11

kkk

2111

2

kk

2

2

1

12

kk

kk

Proof:

2222222 111

3

1

2

11

1

1

3

1

2

11

kkk

2111

2

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

Proof:

2222222 111

3

1

2

11

1

1

3

1

2

11

kkk

2111

2

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

222

1

1

12

kkkk

kk

Proof:

2222222 111

3

1

2

11

1

1

3

1

2

11

kkk

2111

2

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

222

1

1

12

kkkk

kk

21

12

kk

kk

Proof:

2222222 111

3

1

2

11

1

1

3

1

2

11

kkk

2111

2

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

222

1

1

12

kkkk

kk

21

12

kk

kk

1

12

k

Proof:

2222222 111

3

1

2

11

1

1

3

1

2

11

kkk

2111

2

kk

2

2

1

12

kk

kk

22

1

12

kk

kk

222

1

1

12

kkkk

kk

21

12

kk

kk

1

12

k

11

21

1

3

1

2

11

222

kk

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

1for 2 2 11 naaa nn

(ii) A sequence is defined by;

1for 2 that Show nan

Test: n = 1 221 a

2 kaknA

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

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:

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

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

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

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

nn

nnn

nnn

yx

yxy

yxxyx

2,

2 2,5 1111

1for 10 Prove nyx nn

by; defined are and sequences The nn yxiii

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

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

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

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

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

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

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

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

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

62.1

2

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

62.1

2

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

62.1

2

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

62.1

2

51..

1

SHR

SHRSHL ....

1

.. 2

aSHL

62.2

2

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

62.1

2

51..

1

SHR

SHRSHL ....

1

.. 2

aSHL

62.2

2

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

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 ....

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 ....

Proof: 11 kkk aaa

Proof: 11 kkk aaa1

2

51

2

51

kk

Proof: 11 kkk aaa1

2

51

2

51

kk

211

2

51

2

51

2

51k

Proof: 11 kkk aaa1

2

51

2

51

kk

211

2

51

2

51

2

51k

2

1

51

4

51

2

2

51k

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

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

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

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*