11x1 t10 08 mathematical induction 1
TRANSCRIPT
Mathematical Induction
Mathematical InductionStep 1: Prove the result is true for n = 1 (or whatever the first term
is)
Mathematical InductionStep 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 InductionStep 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 InductionStep 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 + 1NOTE: 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 InductionStep 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 + 1NOTE: It is important to note in your conclusion that the result is
true for n = k + 1 if it is true for n = kStep 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction
12123112531 .. 2222 nnnnige
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
RHS
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
RHS
RHSLHS
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
RHS
RHSLHS Hence the result is true for n = 1
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
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
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
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
12123112531 .. 2222 kkkkei
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
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
12123112531 .. 2222 kkkkei
Step 3: Prove the result is true for n = k + 1
12123112531 .. 2222 nnnnige
Step 1: Prove the result is true for n = 1
112
LHS
1
31131
1212131
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
12123112531 .. 2222 kkkkei
Step 3: Prove the result is true for n = k + 1
321213112531 :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
212121231
kkkk
kS
1
kT
Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
123121231
kkkk
Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
123121231
kkkk
3521231
3621231
2
2
kkk
kkkk
Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
123121231
kkkk
3521231
3621231
2
2
kkk
kkkk
3211231
kkk
Proof: 22222
2222
1212531
12531
kk
k
212121231
kkkk
kS
1
kT
12123112 kkkk
123121231
kkkk
3521231
3621231
2
2
kkk
kkkk
3211231
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 nn
kkii
1 1212121
n
k nn
kkii
1 1212121
1212121
751
531
311
nn
nn
n
k nn
kkii
1 1212121
Step 1: Prove the result is true for n = 1 121212
175
153
131
1
n
nnn
n
k nn
kkii
1 1212121
Step 1: Prove the result is true for n = 1
31
311
LHS
1212121
751
531
311
nn
nn
n
k nn
kkii
1 1212121
Step 1: Prove the result is true for n = 1
31
311
LHS
31
121
RHS
1212121
751
531
311
nn
nn
n
k nn
kkii
1 1212121
Step 1: Prove the result is true for n = 1
31
311
LHS
31
121
RHS
RHSLHS
1212121
751
531
311
nn
nn
n
k nn
kkii
1 1212121
Step 1: Prove the result is true for n = 1
31
311
LHS
31
121
RHS
RHSLHS Hence the result is true for n = 1
1212121
751
531
311
nn
nn
n
k nn
kkii
1 1212121
Step 1: Prove the result is true for n = 1
31
311
LHS
31
121
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
1212121
751
531
311
nn
nn
n
k nn
kkii
1 1212121
Step 1: Prove the result is true for n = 1
31
311
LHS
31
121
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
1212121
751
531
311
nn
nn
1212121
751
531
311..
kk
kkei
Step 3: Prove the result is true for n = k + 1
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
Proof:
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
Proof:
32121
12121
751
531
311
32121
751
531
311
kkkk
kk
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
Proof:
32121
12121
751
531
311
32121
751
531
311
kkkk
kk
32121
12
kkkk
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
Proof:
32121
12121
751
531
311
32121
751
531
311
kkkk
kk
32121
12
kkkk
3212
132
kk
kk
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
Proof:
32121
12121
751
531
311
32121
751
531
311
kkkk
kk
32121
12
kkkk
3212
132
kk
kk
3212132 2
kkkk
Step 3: Prove the result is true for n = k + 1
321
32121
751
531
311 :Prove ..
k
kkk
ei
Proof:
32121
12121
751
531
311
32121
751
531
311
kkkk
kk
32121
12
kkkk
3212
132
kk
kk
3212132 2
kkkk
3212
112
kkkk
32
1
kk
32
1
kk
Hence the result is true for n = k + 1 if it is also true for n = k
32
1
kk
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
kk
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