mathematical induction
-
Upload
ankushkumar -
Category
Documents
-
view
232 -
download
0
description
Transcript of mathematical induction
![Page 1: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/1.jpg)
Mathematical Induction
![Page 2: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/2.jpg)
Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term
is)
![Page 3: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/3.jpg)
Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term
is)
Step 2: Assume the result is true for n = k, where k is a positive
integer (or another condition that matches the question)
![Page 4: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/4.jpg)
Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term
is)
Step 2: Assume the result is true for n = k, where k is a positive
integer (or another condition that matches the question)
Step 3: Prove the result is true for n = k + 1
![Page 5: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/5.jpg)
Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term
is)
Step 2: Assume the result is true for n = k, where k is a positive
integer (or another condition that matches the question)
Step 3: Prove the result is true for n = k + 1
NOTE: It is important to note in your conclusion that the result is
true for n = k + 1 if it is true for n = k
![Page 6: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/6.jpg)
Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term
is)
Step 2: Assume the result is true for n = k, where k is a positive
integer (or another condition that matches the question)
Step 3: Prove the result is true for n = k + 1
NOTE: It is important to note in your conclusion that the result is
true for n = k + 1 if it is true for n = k
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction
![Page 7: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/7.jpg)
12123
112531 ..
2222 nnnnige
![Page 8: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/8.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
![Page 9: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/9.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
![Page 10: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/10.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
![Page 11: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/11.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
RHSLHS
![Page 12: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/12.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
RHSLHS
Hence the result is true for n = 1
![Page 13: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/13.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
RHSLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
![Page 14: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/14.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
RHSLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
12123
112531 ..
2222 kkkkei
![Page 15: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/15.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
RHSLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
12123
112531 ..
2222 kkkkei
Step 3: Prove the result is true for n = k + 1
![Page 16: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/16.jpg)
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
RHSLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
12123
112531 ..
2222 kkkkei
Step 3: Prove the result is true for n = k + 1
321213
112531 :Prove ..
2222 kkkkei
![Page 17: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/17.jpg)
Proof:
![Page 18: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/18.jpg)
Proof:
22222
2222
1212531
12531
kk
k
![Page 19: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/19.jpg)
Proof:
22222
2222
1212531
12531
kk
k
kS
![Page 20: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/20.jpg)
Proof:
22222
2222
1212531
12531
kk
k
kS
1
kT
![Page 21: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/21.jpg)
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
![Page 22: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/22.jpg)
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
1212
3
112 kkkk
![Page 23: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/23.jpg)
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
1212
3
112 kkkk
12312123
1 kkkk
![Page 24: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/24.jpg)
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
1212
3
112 kkkk
12312123
1 kkkk
352123
1
362123
1
2
2
kkk
kkkk
![Page 25: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/25.jpg)
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
1212
3
112 kkkk
12312123
1 kkkk
352123
1
362123
1
2
2
kkk
kkkk
321123
1 kkk
![Page 26: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/26.jpg)
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
1212
3
112 kkkk
12312123
1 kkkk
352123
1
362123
1
2
2
kkk
kkkk
321123
1 kkk
Hence the result is true for n = k + 1 if it is also true for n = k
![Page 27: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/27.jpg)
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction
![Page 28: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/28.jpg)
n
k n
n
kkii
1 121212
1
![Page 29: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/29.jpg)
n
k n
n
kkii
1 121212
1
121212
1
75
1
53
1
31
1
n
n
nn
![Page 30: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/30.jpg)
n
k n
n
kkii
1 121212
1
Step 1: Prove the result is true for n = 1
121212
1
75
1
53
1
31
1
n
n
nn
![Page 31: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/31.jpg)
n
k n
n
kkii
1 121212
1
Step 1: Prove the result is true for n = 1
3
1
31
1
LHS
121212
1
75
1
53
1
31
1
n
n
nn
![Page 32: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/32.jpg)
n
k n
n
kkii
1 121212
1
Step 1: Prove the result is true for n = 1
3
1
31
1
LHS
3
1
12
1
RHS
121212
1
75
1
53
1
31
1
n
n
nn
![Page 33: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/33.jpg)
n
k n
n
kkii
1 121212
1
Step 1: Prove the result is true for n = 1
3
1
31
1
LHS
3
1
12
1
RHS
RHSLHS
121212
1
75
1
53
1
31
1
n
n
nn
![Page 34: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/34.jpg)
n
k n
n
kkii
1 121212
1
Step 1: Prove the result is true for n = 1
3
1
31
1
LHS
3
1
12
1
RHS
RHSLHS
Hence the result is true for n = 1
121212
1
75
1
53
1
31
1
n
n
nn
![Page 35: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/35.jpg)
n
k n
n
kkii
1 121212
1
Step 1: Prove the result is true for n = 1
3
1
31
1
LHS
3
1
12
1
RHS
RHSLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
121212
1
75
1
53
1
31
1
n
n
nn
![Page 36: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/36.jpg)
n
k n
n
kkii
1 121212
1
Step 1: Prove the result is true for n = 1
3
1
31
1
LHS
3
1
12
1
RHS
RHSLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
121212
1
75
1
53
1
31
1
n
n
nn
121212
1
75
1
53
1
31
1..
k
k
kkei
![Page 37: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/37.jpg)
Step 3: Prove the result is true for n = k + 1
![Page 38: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/38.jpg)
Step 3: Prove the result is true for n = k + 1
32
1
3212
1
75
1
53
1
31
1 :Prove ..
k
k
kkei
![Page 39: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/39.jpg)
Step 3: Prove the result is true for n = k + 1
32
1
3212
1
75
1
53
1
31
1 :Prove ..
k
k
kkei
Proof:
![Page 40: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/40.jpg)
Step 3: Prove the result is true for n = k + 1
32
1
3212
1
75
1
53
1
31
1 :Prove ..
k
k
kkei
Proof:
3212
1
1212
1
75
1
53
1
31
1
3212
1
75
1
53
1
31
1
kkkk
kk
![Page 41: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/41.jpg)
Step 3: Prove the result is true for n = k + 1
32
1
3212
1
75
1
53
1
31
1 :Prove ..
k
k
kkei
Proof:
3212
1
1212
1
75
1
53
1
31
1
3212
1
75
1
53
1
31
1
kkkk
kk
3212
1
12
kkk
k
![Page 42: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/42.jpg)
Step 3: Prove the result is true for n = k + 1
32
1
3212
1
75
1
53
1
31
1 :Prove ..
k
k
kkei
Proof:
3212
1
1212
1
75
1
53
1
31
1
3212
1
75
1
53
1
31
1
kkkk
kk
3212
1
12
kkk
k
3212
132
kk
kk
![Page 43: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/43.jpg)
Step 3: Prove the result is true for n = k + 1
32
1
3212
1
75
1
53
1
31
1 :Prove ..
k
k
kkei
Proof:
3212
1
1212
1
75
1
53
1
31
1
3212
1
75
1
53
1
31
1
kkkk
kk
3212
1
12
kkk
k
3212
132
kk
kk
3212
132 2
kk
kk
![Page 44: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/44.jpg)
Step 3: Prove the result is true for n = k + 1
32
1
3212
1
75
1
53
1
31
1 :Prove ..
k
k
kkei
Proof:
3212
1
1212
1
75
1
53
1
31
1
3212
1
75
1
53
1
31
1
kkkk
kk
3212
1
12
kkk
k
3212
132
kk
kk
3212
132 2
kk
kk
3212
112
kk
kk
![Page 45: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/45.jpg)
32
1
k
k
![Page 46: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/46.jpg)
32
1
k
k
Hence the result is true for n = k + 1 if it is also true for n = k
![Page 47: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/47.jpg)
32
1
k
k
Hence the result is true for n = k + 1 if it is also true for n = k
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction
![Page 48: mathematical induction](https://reader034.fdocuments.net/reader034/viewer/2022042601/5469ecfcaf7959e8488b4645/html5/thumbnails/48.jpg)
32
1
k
k
Hence the result is true for n = k + 1 if it is also true for n = k
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction
Exercise 6N; 1 ace etc, 10(polygon), 13