x2 t08 02 induction
TRANSCRIPT
![Page 1: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/1.jpg)
Mathematical Induction
![Page 2: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/2.jpg)
Mathematical Induction
nni 121
31
211 Prove e.g. 222
![Page 3: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/3.jpg)
Mathematical Induction
nni 121
31
211 Prove e.g. 222
Test: n = 1
![Page 4: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/4.jpg)
Mathematical Induction
nni 121
31
211 Prove e.g. 222
Test: n = 1
111.. 2
SHL
![Page 5: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/5.jpg)
Mathematical Induction
nni 121
31
211 Prove e.g. 222
Test: n = 1
111.. 2
SHL
1112..
SHR
![Page 6: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/6.jpg)
Mathematical Induction
nni 121
31
211 Prove e.g. 222
Test: n = 1
111.. 2
SHL
1112..
SHR
SHRSHL ....
![Page 7: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/7.jpg)
Mathematical Induction
nni 121
31
211 Prove e.g. 222
Test: n = 1
111.. 2
SHL
1112..
SHR
SHRSHL ....
kk
knA 12131
211 222
![Page 8: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/8.jpg)
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
![Page 9: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/9.jpg)
Proof:
2222222 111
31
211
11
31
211
kkk
![Page 10: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/10.jpg)
Proof:
2222222 111
31
211
11
31
211
kkk
21112
kk
![Page 11: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/11.jpg)
Proof:
2222222 111
31
211
11
31
211
kkk
21112
kk
2
2
112
kk
kk
![Page 12: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/12.jpg)
Proof:
2222222 111
31
211
11
31
211
kkk
21112
kk
2
2
112
kk
kk
22
112
kk
kk
![Page 13: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/13.jpg)
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
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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
![Page 15: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/15.jpg)
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
![Page 16: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/16.jpg)
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
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1for 2 2 11 naaa nn
(ii) A sequence is defined by;
1for 2that Show nan
![Page 18: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/18.jpg)
1for 2 2 11 naaa nn
(ii) A sequence is defined by;
1for 2that Show nan
Test: n = 1
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1for 2 2 11 naaa nn
(ii) A sequence is defined by;
1for 2that Show nan
Test: n = 1 221 a
![Page 20: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/20.jpg)
1for 2 2 11 naaa nn
(ii) A sequence is defined by;
1for 2that Show nan
Test: n = 1 221 a
2 kaknA
![Page 21: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/21.jpg)
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
![Page 22: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/22.jpg)
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:
![Page 23: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/23.jpg)
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
![Page 24: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/24.jpg)
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
![Page 25: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/25.jpg)
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
![Page 26: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/26.jpg)
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
![Page 27: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/27.jpg)
nn
nnn
nnn yx
yxyyxxyx
2,
2 2,5 1111
1for 10 Prove nyx nn
by;definedareandsequencesThe nn yxiii
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nn
nnn
nnn yx
yxyyxxyx
2,
2 2,5 1111
1for 10 Prove nyx nn
Test: n = 1
by;definedareandsequencesThe nn yxiii
![Page 29: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/29.jpg)
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
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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
![Page 31: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/31.jpg)
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
![Page 32: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/32.jpg)
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
![Page 33: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/33.jpg)
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
![Page 34: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/34.jpg)
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
![Page 35: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/35.jpg)
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
![Page 36: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/36.jpg)
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](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/37.jpg)
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](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/38.jpg)
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](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/39.jpg)
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
![Page 40: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/40.jpg)
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 ....
![Page 41: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/41.jpg)
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
![Page 42: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/42.jpg)
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
![Page 43: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/43.jpg)
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 ....
![Page 44: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/44.jpg)
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 ....
![Page 45: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/45.jpg)
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 ....
![Page 46: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/46.jpg)
Proof: 11 kkk aaa
![Page 47: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/47.jpg)
Proof: 11 kkk aaa1
251
251
kk
![Page 48: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/48.jpg)
Proof: 11 kkk aaa1
251
251
kk
211
251
251
251
k
![Page 49: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/49.jpg)
Proof: 11 kkk aaa1
251
251
kk
211
251
251
251
k
2
1
514
512
251
k
![Page 50: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/50.jpg)
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
![Page 51: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/51.jpg)
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
![Page 52: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/52.jpg)
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
![Page 53: X2 T08 02 induction](https://reader034.vdocuments.site/reader034/viewer/2022042607/55b60317bb61eb170a8b479c/html5/thumbnails/53.jpg)
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*