x2 t08 02 induction
TRANSCRIPT
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*