Download - 11 x1 t14 10 mathematical induction 3 (2012)
Mathematical Induction
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
25
52
RHS
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
25
52
RHS
RHSLHS
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
Hence the result is true for n = 5
25
52
RHS
RHSLHS
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
Hence the result is true for n = 5
Step 2: Assume the result is true for n = k, where k is a positive
integer > 4 22 .. kei k
25
52
RHS
RHSLHS
Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
32
25
LHS
Hence the result is true for n = 5
Step 2: Assume the result is true for n = k, where k is a positive
integer > 4 22 .. kei k
Step 3: Prove the result is true for n = k + 1
21 12 :Prove .. kei k
25
52
RHS
RHSLHS
Proof:
Proof: 12 k
Proof: 12 k k22
Proof: 12 k k22
22k
Proof: 12 k k22
22k22 kk
Proof: 12 k k22
22k22 kk
kkk 2
Proof: 12 k k22
22k22 kk
kkk 2
kk 42
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
Proof: 12 k k22
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
21 12 kk
Proof: 12 k k22
Hence the result is true for n = k + 1 if it is also true for n = k
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
21 12 kk
Proof: 12 k k22
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 = 5, then the result is true for
all positive integral values of n > 4 by induction .
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
21 12 kk
Proof: 12 k k22
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 = 5, then the result is true for
all positive integral values of n > 4 by induction .
22k22 kk
kkk 2
kk 42 4k
kkk 222
822 kk 4k
122 kk
21 k
21 12 kk
Exercise 6N;
6 abc, 8a, 15