11 x1 t14 08 mathematical induction 1 (2013)
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Transcript of 11 x1 t14 08 mathematical induction 1 (2013)
Mathematical Induction
Mathematical Induction Step 1: Prove the result is true for n = 1 (or whatever the first term
is)
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)
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
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
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
12123
112531 ..
2222 nnnnige
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
12123
112531 ..
2222 nnnnige
Step 1: Prove the result is true for n = 1
1
12
LHS
1
3113
1
121213
1
RHS
RHSLHS
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
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 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
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
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
Proof:
Proof:
22222
2222
1212531
12531
kk
k
Proof:
22222
2222
1212531
12531
kk
k
kS
Proof:
22222
2222
1212531
12531
kk
k
kS
1
kT
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
1212
3
112 kkkk
Proof:
22222
2222
1212531
12531
kk
k
21212123
1 kkkk
kS
1
kT
1212
3
112 kkkk
12312123
1 kkkk
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
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
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
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction
n
k n
n
kkii
1 121212
1
n
k n
n
kkii
1 121212
1
121212
1
75
1
53
1
31
1
n
n
nn
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
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
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
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
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
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
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
Step 3: Prove the result is true for n = k + 1
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
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:
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
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
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
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
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
32
1
k
k
32
1
k
k
Hence the result is true for n = k + 1 if it is also true for n = k
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
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