11x1 t10 10 mathematical induction 3
TRANSCRIPT
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Mathematical Induction
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
3225
LHS
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
3225
LHS
2552
RHS
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
3225
LHS
2552
RHS
RHSLHS
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
3225
LHS
Hence the result is true for n = 5
2552
RHS
RHSLHS
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
3225
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
2552
RHS
RHSLHS
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Mathematical Induction 4for 2 Prove .. 2 nnvge n
Step 1: Prove the result is true for n = 5
3225
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
2552
RHS
RHSLHS
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Proof:
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Proof: 12 k
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Proof: 12 k k22
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Proof: 12 k k22 22k
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Proof: 12 k k22 22k
22 kk
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Proof: 12 k k22 22k
22 kk kkk 2
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Proof: 12 k k22 22k
22 kk kkk 2
kk 42
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Proof: 12 k k22 22k
22 kk kkk 2
kk 42 4k
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Proof: 12 k k22 22k
22 kk kkk 2
kk 42 4kkkk 222
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Proof: 12 k k22 22k
22 kk kkk 2
kk 42 4kkkk 222
822 kk
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Proof: 12 k k22 22k
22 kk kkk 2
kk 42 4kkkk 222
822 kk 4k
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Proof: 12 k k22 22k
22 kk kkk 2
kk 42 4kkkk 222
822 kk 4k122 kk
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Proof: 12 k k22 22k
22 kk kkk 2
kk 42 4kkkk 222
822 kk 4k122 kk
21 k
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Proof: 12 k k22 22k
22 kk kkk 2
kk 42 4kkkk 222
822 kk 4k122 kk
21 k 21 12 kk
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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 4kkkk 222
822 kk 4k122 kk
21 k 21 12 kk
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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 4kkkk 222
822 kk 4k122 kk
21 k 21 12 kk
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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 4kkkk 222
822 kk 4k122 kk
21 k 21 12 kk
Exercise 6N; 6 abc, 8a, 15