11 x1 t14 09 mathematical induction 2 (2013)
TRANSCRIPT
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Mathematical Induction
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
which is divisible by 3
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
Step 3: Prove the result is true for n = k + 1
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Mathematical Induction 3by divisible is 21 Prove .. nnniiige
Step 1: Prove the result is true for n = 1
6
321
Hence the result is true for n = 1
which is divisible by 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,321 .. PPkkkei
Step 3: Prove the result is true for n = k + 1
integeran is where,3321 :Prove .. QQkkkei
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Proof:
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Proof: 321 kkk
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Proof: 321 kkk
21321 kkkkk
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Proof: 321 kkk
21321 kkkkk
2133 kkP
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Proof: 321 kkk
21321 kkkkk
2133 kkP
213 kkP
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Proof: 321 kkk
21321 kkkkk
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
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Proof: 321 kkk
21321 kkkkk
Hence the result is true for n = k + 1 if it is also true for n = k
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
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Proof: 321 kkk
21321 kkkkk
Hence the result is true for n = k + 1 if it is also true for n = k
2133 kkP
213 kkP
integeran is which 21 where,3 kkPQQ
Step 4: Since the result is true for n = 1, then the result is true for
all positive integral values of n by induction.
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5by divisible is 23 Prove 23 nniv
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5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
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5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
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5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
which is divisible by 5
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5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
which is divisible by 5
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5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,523 .. 23 PPei kk
which is divisible by 5
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5by divisible is 23 Prove 23 nniv
Step 1: Prove the result is true for n = 1
35
827
23 33
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
integeran is where,523 .. 23 PPei kk
which is divisible by 5
Step 3: Prove the result is true for n = k + 1
integeran is where,523 :Prove .. 333 QQei kk
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Proof:
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Proof: 333 23 kk
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Proof: 333 23 kk
33 2327 kk
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Proof: 333 23 kk
33 2327 kk
32 22527 kkP
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Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
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Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP
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Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
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Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
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Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
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Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
Hence the result is true for n = k + 1 if it is also true for n = k
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Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
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 37: 11 x1 t14 09 mathematical induction 2 (2013)](https://reader035.vdocuments.us/reader035/viewer/2022062406/558e0e5d1a28ab64318b4781/html5/thumbnails/37.jpg)
Proof: 333 23 kk
33 2327 kk
32 22527 kkP32 2227135 kkP
22 22227135 kkP2225135 kP
225275 kP
integeran is which 2527 where,5 2 kPQQ
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;
3bdf, 4 ab(i), 9